Comprehensive MCQ Set on Functions and Graphs: Analysis,
Transformations, and Graphing Techniques
1. What is the domain of the function f(x) = √(x + 2)?
a) (-∞, -2]
b) [-2, ∞)
c) (-2, ∞)
d) [-2, ∞)
Answer: b) [-2, ∞)
Solution: The expression under the square root must be non-negative.
x + 2 ≥ 0
x ≥ -2
Therefore, the domain is [-2, ∞)
2. Which transformation represents y = f(x - 3)?
a) Shift 3 units right
b) Shift 3 units left
c) Shift 3 units up
d) Shift 3 units down
Answer: a) Shift 3 units right
Solution: When subtracting from x inside the function, the graph shifts to the right by that amount.
3. What is the range of f(x) = |x - 2|?
a) [0, ∞)
b) (-∞, 0]
c) (-∞, ∞)
d) [2, ∞)
Answer: a) [0, ∞)
Solution: The absolute value function always returns non-negative values, and the minimum value
occurs when x - 2 = 0, or x = 2.
4. Which function represents a vertical stretch by a factor of 2 of f(x)?
a) 2f(x)
b) f(2x)
c) f(x/2)
d) f(x) + 2
Answer: a) 2f(x)
Solution: Multiplying the entire function by a constant greater than 1 stretches the graph vertically.
5. What is the period of y = sin(2x)?
a) π
b) 2π
c) π/2
d) 4π
Answer: c) π/2
Solution: The period of sin(kx) is (2π)/k. Here, k = 2, so the period is 2π/2 = π/2.
6. Which of the following is an even function?
a) f(x) = x³
b) f(x) = x²
c) f(x) = x
d) f(x) = |x| - 1
Answer: b) f(x) = x²
Solution: For an even function, f(-x) = f(x). This is true for x², as (-x)² = x².
7. What is the x-intercept of f(x) = 3x - 6?
a) 2
b) -2
c) 6
d) 3
Answer: a) 2
Solution: Set f(x) = 0 and solve:
3x - 6 = 0
3x = 6
x = 2
8. Which transformation represents y = -f(x)?
a) Reflection over x-axis
b) Reflection over y-axis
c) Vertical stretch by factor of -1
d) Both a and c
Answer: d) Both a and c
Solution: Multiplying f(x) by -1 both reflects the graph over the x-axis and stretches it vertically by a
factor of -1.
9. What is the slope of the line passing through (2, 5) and (4, 9)?
a) 2
b) 4
c) 1
d) 3
Answer: a) 2
Solution: Using the slope formula:
m = (y₂ - y₁) / (x₂ - x₁) = (9 - 5) / (4 - 2) = 4 / 2 = 2
10. Which function represents a horizontal compression by a factor of 3 of f(x)?
a) 3f(x)
b) f(3x)
c) f(x/3)
d) f(x) + 3
Answer: b) f(3x)
Solution: Multiplying x by a constant greater than 1 inside the function compresses the graph
horizontally.
11. What is the vertex of the parabola y = (x - 2)² - 3?
a) (2, -3)
b) (-2, -3)
c) (2, 3)
d) (-2, 3)
Answer: a) (2, -3)
Solution: For y = (x - h)² + k, the vertex is (h, k). Here, h = 2 and k = -3.
12. Which of the following represents an inverse function of f(x) = 2x + 3?
a) f⁻¹(x) = (x - 3)/2
b) f⁻¹(x) = 2x - 3
c) f⁻¹(x) = (x + 3)/2
d) f⁻¹(x) = -2x + 3
Answer: a) f⁻¹(x) = (x - 3)/2
Solution: To find the inverse, swap x and y, then solve for y:
y = 2x + 3
x = 2y + 3
x - 3 = 2y
(x - 3)/2 = y
13. What is the domain of f(x) = 1 / (x² - 4)?
a) All real numbers
b) All real numbers except 2 and -2
c) All real numbers except 0
d) All positive real numbers
Answer: b) All real numbers except 2 and -2
Solution: The denominator cannot be zero:
x² - 4 ≠ 0
(x + 2)(x - 2) ≠ 0
x ≠ 2 and x ≠ -2
14. Which of the following is an odd function?
a) f(x) = x⁴
b) f(x) = x³
c) f(x) = x²
d) f(x) = |x|
Answer: b) f(x) = x³
Solution: For an odd function, f(-x) = -f(x). This is true for x³, as (-x)³ = -x³.
15. What is the range of f(x) = √x?
a) [0, ∞)
b) (-∞, 0]
c) (-∞, ∞)
d) [0, 1]
Answer: a) [0, ∞)
Solution: The square root of a non-negative number is always non-negative, and there's no upper
limit.
16. Which transformation represents y = f(x) + 3?
a) Shift 3 units right
b) Shift 3 units left
c) Shift 3 units up
d) Shift 3 units down
Answer: c) Shift 3 units up
Solution: Adding a constant to the entire function shifts the graph vertically upward by that
amount.
17. What is the period of y = cos(x/2)?
a) π
b) 2π
c) 4π
d) π/2
Answer: c) 4π
Solution: The period of cos(kx) is (2π)/k. Here, k = 1/2, so the period is 2π/(1/2) = 4π.
18. Which function represents a reflection of f(x) over the y-axis?
a) -f(x)
b) f(-x)
c) -f(-x)
d) f(x + 1)
Answer: b) f(-x)
Solution: Negating x inside the function reflects the graph over the y-axis.
19. What is the y-intercept of f(x) = 2x² - 3x + 5?
a) 2
b) -3
c) 5
d) 0
Answer: c) 5
Solution: The y-intercept is f(0):
f(0) = 2(0)² - 3(0) + 5 = 5
20. Which of the following represents a vertical asymptote?
a) lim(x→a) f(x) = ∞
b) lim(x→∞) f(x) = b
c) f(x) = 0
d) f'(x) = 0
Answer: a) lim(x→a) f(x) = ∞
Solution: A vertical asymptote occurs when the function approaches infinity as x approaches a
finite value.
21. What is the slope of the line 3x - 2y = 6?
a) 3/2
b) -3/2
c) 2/3
d) -2/3
Answer: a) 3/2
Solution: Rearrange to slope-intercept form:
-2y = -3x + 6
y = (3/2)x - 3
The coefficient of x is the slope.
22. Which function represents a horizontal reflection and vertical stretch by factor 2 of f(x)?
a) 2f(x)
b) f(2x)
c) 2f(-x)
d) -2f(x)
Answer: c) 2f(-x)
Solution: f(-x) reflects over y-axis, and multiplying by 2 stretches vertically.
23. What is the domain of f(x) = log₂(x - 3)?
a) (3, ∞)
b) [3, ∞)
c) (-∞, 3)
d) (-∞, 3]
Answer: a) (3, ∞)
Solution: The argument of log must be positive:
x - 3 > 0
x > 3
24. Which of the following represents a horizontal asymptote?
a) lim(x→a) f(x) = ∞
b) lim(x→∞) f(x) = b
c) f(x) = 0
d) f'(x) = 0
Answer: b) lim(x→∞) f(x) = b
Solution: A horizontal asymptote occurs when the function approaches a finite value as x
approaches infinity.
25. What is the range of f(x) = 1 / (x² + 1)?
a) (0, 1]
b) [0, 1)
c) (0, ∞)
d) (-∞, ∞)
Answer: a) (0, 1]
Solution: The denominator is always positive and at least 1, so 0 < f(x) ≤ 1, with f(x) = 1 when x = 0.
26. Which transformation represents y = f(2x + 1)?
a) Shift 1 unit left and stretch horizontally by factor 1/2
b) Shift 1 unit right and stretch horizontally by factor 1/2
c) Shift 1/2 unit left and compress horizontally by factor 2
d) Shift 1/2 unit right and compress horizontally by factor 2
Answer: c) Shift 1/2 unit left and compress horizontally by factor 2
Solution: First, factor out the 2: f(2(x + 1/2)). This shows a horizontal compression by 2 and a shift
left by 1/2.
27. What is the period of y = tan(3x)?
a) π/3
b) π/6
c) 2π/3
d) π
Answer: a) π/3
Solution: The period of tan(kx) is π/k. Here, k = 3, so the period is π/3.
28. Which of the following is neither even nor odd?
a) f(x) = x⁴
b) f(x) = x³
c) f(x) = x² + x
d) f(x) = |x|
Answer: c) f(x) = x² + x
Solution: For this function, f(-x) ≠ f(x) and f(-x) ≠ -f(x), so it's neither even nor odd.
29. What is the x-intercept of f(x) = e^x - 4?
a) ln(4)
b) e⁴
c) 4
d) 0
Answer: a) ln(4)
Solution: Set f(x) = 0 and solve:
e^x - 4 = 0
e^x = 4
x = ln(4)
30. Which function represents a vertical reflection and horizontal stretch by factor 2 of f(x)?
a) -f(x/2)
b) f(-x/2)
c) -f(2x)
d) f(-2x)
Answer: a) -f(x/2)
Solution: -f(x) reflects over x-axis, and f(x/2) stretches horizontally by factor 2.
31. What is the domain of f(x) = arcsin(x)?
a) [-1, 1]
b) (-1, 1)
c) [0, π]
d) (-π/2, π/2)
Answer: a) [-1, 1]
Solution: The arcsine function is defined only for inputs between -1 and 1, inclusive.
32. Which of the following represents an oblique asymptote?
a) lim(x→∞) f(x) = mx + b, where m ≠ 0
b) lim(x→a) f(x) = ∞
c) lim(x→∞) f(x) = b
d) f'(x) = 0
Answer: a) lim(x→∞) f(x) = mx + b, where m ≠ 0
Solution: An oblique asymptote is a slant line that the function approaches as x approaches
infinity.
33. What is the range of f(x) = 3 - e^x?
a) (3, ∞)
b) (-∞, 3)
c) (0, 3)
d) (-∞, ∞)
Answer: b) (-∞, 3)
Solution: As x increases, e^x increases without bound, so 3 - e^x decreases without bound. The
upper limit is 3 (as x approaches -∞).
34. Which transformation represents y = -2f(x - 1) + 3?
a) Reflect over x-axis, stretch vertically by 2, shift right 1, shift up 3
b) Reflect over y-axis, stretch vertically by 2, shift left 1, shift up 3
c) Reflect over x-axis, stretch vertically by 2, shift left 1, shift up 3
d) Reflect over y-axis, compress vertically by 1/2, shift right 1, shift down 3
Answer: c) Reflect over x-axis, stretch vertically by 2, shift left 1, shift up 3
Solution: -2f(x) reflects over x-axis and stretches vertically by 2, (x-1) shifts right by 1, and +3 shifts
up by 3.
8. Which transformation represents y = -f(x)?
a) Reflection over x-axis
b) Reflection over y-axis
c) Vertical stretch by factor of -1
d) Both a and c
Answer: d) Both a and c
Solution: Multiplying f(x) by -1 both reflects the graph over the x-axis and stretches it vertically by a
factor of -1.
9. What is the slope of the line passing through (2, 5) and (4, 9)?
a) 2
b) 4
c) 1
d) 3
Answer: a) 2
Solution: Using the slope formula:
m = (y₂ - y₁) / (x₂ - x₁) = (9 - 5) / (4 - 2) = 4 / 2 = 2
10. Which function represents a horizontal compression by a factor of 3 of f(x)?
a) 3f(x)
b) f(3x)
c) f(x/3)
d) f(x) + 3
Answer: b) f(3x)
Solution: Multiplying x by a constant greater than 1 inside the function compresses the graph
horizontally.
11. What is the vertex of the parabola y = (x - 2)² - 3?
a) (2, -3)
b) (-2, -3)
c) (2, 3)
d) (-2, 3)
Answer: a) (2, -3)
Solution: For y = (x - h)² + k, the vertex is (h, k). Here, h = 2 and k = -3.
12. Which of the following represents an inverse function of f(x) = 2x + 3?
a) f⁻¹(x) = (x - 3)/2
b) f⁻¹(x) = 2x - 3
c) f⁻¹(x) = (x + 3)/2
d) f⁻¹(x) = -2x + 3
Answer: a) f⁻¹(x) = (x - 3)/2
Solution: To find the inverse, swap x and y, then solve for y:
y = 2x + 3
x = 2y + 3
x - 3 = 2y
(x - 3)/2 = y
13. What is the domain of f(x) = 1 / (x² - 4)?
a) All real numbers
b) All real numbers except 2 and -2
c) All real numbers except 0
d) All positive real numbers
Answer: b) All real numbers except 2 and -2
Solution: The denominator cannot be zero:
x² - 4 ≠ 0
(x + 2)(x - 2) ≠ 0
x ≠ 2 and x ≠ -2
14. Which of the following is an odd function?
a) f(x) = x⁴
b) f(x) = x³
c) f(x) = x²
d) f(x) = |x|
Answer: b) f(x) = x³
Solution: For an odd function, f(-x) = -f(x). This is true for x³, as (-x)³ = -x³.
15. What is the range of f(x) = √x?
a) [0, ∞)
b) (-∞, 0]
c) (-∞, ∞)
d) [0, 1]
Answer: a) [0, ∞)
Solution: The square root of a non-negative number is always non-negative, and there's no upper
limit.
16. Which transformation represents y = f(x) + 3?
a) Shift 3 units right
b) Shift 3 units left
c) Shift 3 units up
d) Shift 3 units down
Answer: c) Shift 3 units up
Solution: Adding a constant to the entire function shifts the graph vertically upward by that
amount.
17. What is the period of y = cos(x/2)?
a) π
b) 2π
c) 4π
d) π/2
Answer: c) 4π
Solution: The period of cos(kx) is (2π)/k. Here, k = 1/2, so the period is 2π/(1/2) = 4π.
18. Which function represents a reflection of f(x) over the y-axis?
a) -f(x)
b) f(-x)
c) -f(-x)
d) f(x + 1)
Answer: b) f(-x)
Solution: Negating x inside the function reflects the graph over the y-axis.
19. What is the y-intercept of f(x) = 2x² - 3x + 5?
a) 2
b) -3
c) 5
d) 0
Answer: c) 5
Solution: The y-intercept is f(0):
f(0) = 2(0)² - 3(0) + 5 = 5
20. Which of the following represents a vertical asymptote?
a) lim(x→a) f(x) = ∞
b) lim(x→∞) f(x) = b
c) f(x) = 0
d) f'(x) = 0
Answer: a) lim(x→a) f(x) = ∞
Solution: A vertical asymptote occurs when the function approaches infinity as x approaches a
finite value.
21. What is the slope of the line 3x - 2y = 6?
a) 3/2
b) -3/2
c) 2/3
d) -2/3
Answer: a) 3/2
Solution: Rearrange to slope-intercept form:
-2y = -3x + 6
y = (3/2)x - 3
The coefficient of x is the slope.
22. Which function represents a horizontal reflection and vertical stretch by factor 2 of f(x)?
a) 2f(x)
b) f(2x)
c) 2f(-x)
d) -2f(x)
Answer: c) 2f(-x)
Solution: f(-x) reflects over y-axis, and multiplying by 2 stretches vertically.
23. What is the domain of f(x) = log₂(x - 3)?
a) (3, ∞)
b) [3, ∞)
c) (-∞, 3)
d) (-∞, 3]
Answer: a) (3, ∞)
Solution: The argument of log must be positive:
x - 3 > 0
x > 3
24. Which of the following represents a horizontal asymptote?
a) lim(x→a) f(x) = ∞
b) lim(x→∞) f(x) = b
c) f(x) = 0
d) f'(x) = 0
Answer: b) lim(x→∞) f(x) = b
Solution: A horizontal asymptote occurs when the function approaches a finite value as x
approaches infinity.
25. What is the range of f(x) = 1 / (x² + 1)?
a) (0, 1]
b) [0, 1)
c) (0, ∞)
d) (-∞, ∞)
Answer: a) (0, 1]
Solution: The denominator is always positive and at least 1, so 0 < f(x) ≤ 1, with f(x) = 1 when x = 0.
26. Which transformation represents y = f(2x + 1)?
a) Shift 1 unit left and stretch horizontally by factor 1/2
b) Shift 1 unit right and stretch horizontally by factor 1/2
c) Shift 1/2 unit left and compress horizontally by factor 2
d) Shift 1/2 unit right and compress horizontally by factor 2
Answer: c) Shift 1/2 unit left and compress horizontally by factor 2
Solution: First, factor out the 2: f(2(x + 1/2)). This shows a horizontal compression by 2 and a shift
left by 1/2.
27. What is the period of y = tan(3x)?
a) π/3
b) π/6
c) 2π/3
d) π
Answer: a) π/3
Solution: The period of tan(kx) is π/k. Here, k = 3, so the period is π/3.
28. Which of the following is neither even nor odd?
a) f(x) = x⁴
b) f(x) = x³
c) f(x) = x² + x
d) f(x) = |x|
Answer: c) f(x) = x² + x
Solution: For this function, f(-x) ≠ f(x) and f(-x) ≠ -f(x), so it's neither even nor odd.
29. What is the x-intercept of f(x) = e^x - 4?
a) ln(4)
b) e⁴
c) 4
d) 0
Answer: a) ln(4)
Solution: Set f(x) = 0 and solve:
e^x - 4 = 0
e^x = 4
x = ln(4)
30. Which function represents a vertical reflection and horizontal stretch by factor 2 of f(x)?
a) -f(x/2)
b) f(-x/2)
c) -f(2x)
d) f(-2x)
Answer: a) -f(x/2)
Solution: -f(x) reflects over x-axis, and f(x/2) stretches horizontally by factor 2.
31. What is the domain of f(x) = arcsin(x)?
a) [-1, 1]
b) (-1, 1)
c) [0, π]
d) (-π/2, π/2)
Answer: a) [-1, 1]
Solution: The arcsine function is defined only for inputs between -1 and 1, inclusive.
32. Which of the following represents an oblique asymptote?
a) lim(x→∞) f(x) = mx + b, where m ≠ 0
b) lim(x→a) f(x) = ∞
c) lim(x→∞) f(x) = b
d) f'(x) = 0
Answer: a) lim(x→∞) f(x) = mx + b, where m ≠ 0
Solution: An oblique asymptote is a slant line that the function approaches as x approaches
infinity.
33. What is the range of f(x) = 3 - e^x?
a) (3, ∞)
b) (-∞, 3)
c) (0, 3)
d) (-∞, ∞)
Answer: b) (-∞, 3)
Solution: As x increases, e^x increases without bound, so 3 - e^x decreases without bound. The
upper limit is 3 (as x approaches -∞).
34. Which transformation represents y = -2f(x - 1) + 3?
a) Reflect over x-axis, stretch vertically by 2, shift right 1, shift up 3
b) Reflect over y-axis, stretch vertically by 2, shift left 1, shift up 3
c) Reflect over x-axis, stretch vertically by 2, shift left 1, shift up 3
d) Reflect over y-axis, compress vertically by 1/2, shift right 1, shift down 3
Answer: c) Reflect over x-axis, stretch vertically by 2, shift left 1, shift up 3
Solution: -2f(x) reflects over x-axis and stretches vertically by 2, (x-1) shifts right by 1, and +3 shifts
up by 3.
8. Which transformation represents y = -f(x)?
a) Reflection over x-axis
b) Reflection over y-axis
c) Vertical stretch by factor of -1
d) Both a and c
Answer: d) Both a and c
Solution: Multiplying f(x) by -1 both reflects the graph over the x-axis and stretches it vertically by a
factor of -1.
9. What is the slope of the line passing through (2, 5) and (4, 9)?
a) 2
b) 4
c) 1
d) 3
Answer: a) 2
Solution: Using the slope formula:
m = (y₂ - y₁) / (x₂ - x₁) = (9 - 5) / (4 - 2) = 4 / 2 = 2
10. Which function represents a horizontal compression by a factor of 3 of f(x)?
a) 3f(x)
b) f(3x)
c) f(x/3)
d) f(x) + 3
Answer: b) f(3x)
Solution: Multiplying x by a constant greater than 1 inside the function compresses the graph
horizontally.
11. What is the vertex of the parabola y = (x - 2)² - 3?
a) (2, -3)
b) (-2, -3)
c) (2, 3)
d) (-2, 3)
Answer: a) (2, -3)
Solution: For y = (x - h)² + k, the vertex is (h, k). Here, h = 2 and k = -3.
12. Which of the following represents an inverse function of f(x) = 2x + 3?
a) f⁻¹(x) = (x - 3)/2
b) f⁻¹(x) = 2x - 3
c) f⁻¹(x) = (x + 3)/2
d) f⁻¹(x) = -2x + 3
Answer: a) f⁻¹(x) = (x - 3)/2
Solution: To find the inverse, swap x and y, then solve for y:
y = 2x + 3
x = 2y + 3
x - 3 = 2y
(x - 3)/2 = y
13. What is the domain of f(x) = 1 / (x² - 4)?
a) All real numbers
b) All real numbers except 2 and -2
c) All real numbers except 0
d) All positive real numbers
Answer: b) All real numbers except 2 and -2
Solution: The denominator cannot be zero:
x² - 4 ≠ 0
(x + 2)(x - 2) ≠ 0
x ≠ 2 and x ≠ -2
14. Which of the following is an odd function?
a) f(x) = x⁴
b) f(x) = x³
c) f(x) = x²
d) f(x) = |x|
Answer: b) f(x) = x³
Solution: For an odd function, f(-x) = -f(x). This is true for x³, as (-x)³ = -x³.
15. What is the range of f(x) = √x?
a) [0, ∞)
b) (-∞, 0]
c) (-∞, ∞)
d) [0, 1]
Answer: a) [0, ∞)
Solution: The square root of a non-negative number is always non-negative, and there's no upper
limit.
16. Which transformation represents y = f(x) + 3?
a) Shift 3 units right
b) Shift 3 units left
c) Shift 3 units up
d) Shift 3 units down
Answer: c) Shift 3 units up
Solution: Adding a constant to the entire function shifts the graph vertically upward by that
amount.
17. What is the period of y = cos(x/2)?
a) π
b) 2π
c) 4π
d) π/2
Answer: c) 4π
Solution: The period of cos(kx) is (2π)/k. Here, k = 1/2, so the period is 2π/(1/2) = 4π.
18. Which function represents a reflection of f(x) over the y-axis?
a) -f(x)
b) f(-x)
c) -f(-x)
d) f(x + 1)
Answer: b) f(-x)
Solution: Negating x inside the function reflects the graph over the y-axis.
19. What is the y-intercept of f(x) = 2x² - 3x + 5?
a) 2
b) -3
c) 5
d) 0
Answer: c) 5
Solution: The y-intercept is f(0):
f(0) = 2(0)² - 3(0) + 5 = 5
20. Which of the following represents a vertical asymptote?
a) lim(x→a) f(x) = ∞
b) lim(x→∞) f(x) = b
c) f(x) = 0
d) f'(x) = 0
Answer: a) lim(x→a) f(x) = ∞
Solution: A vertical asymptote occurs when the function approaches infinity as x approaches a
finite value.
21. What is the slope of the line 3x - 2y = 6?
a) 3/2
b) -3/2
c) 2/3
d) -2/3
Answer: a) 3/2
Solution: Rearrange to slope-intercept form:
-2y = -3x + 6
y = (3/2)x - 3
The coefficient of x is the slope.
22. Which function represents a horizontal reflection and vertical stretch by factor 2 of f(x)?
a) 2f(x)
b) f(2x)
c) 2f(-x)
d) -2f(x)
Answer: c) 2f(-x)
Solution: f(-x) reflects over y-axis, and multiplying by 2 stretches vertically.
23. What is the domain of f(x) = log₂(x - 3)?
a) (3, ∞)
b) [3, ∞)
c) (-∞, 3)
d) (-∞, 3]
Answer: a) (3, ∞)
Solution: The argument of log must be positive:
x - 3 > 0
x > 3
24. Which of the following represents a horizontal asymptote?
a) lim(x→a) f(x) = ∞
b) lim(x→∞) f(x) = b
c) f(x) = 0
d) f'(x) = 0
Answer: b) lim(x→∞) f(x) = b
Solution: A horizontal asymptote occurs when the function approaches a finite value as x
approaches infinity.
25. What is the range of f(x) = 1 / (x² + 1)?
a) (0, 1]
b) [0, 1)
c) (0, ∞)
d) (-∞, ∞)
Answer: a) (0, 1]
Solution: The denominator is always positive and at least 1, so 0 < f(x) ≤ 1, with f(x) = 1 when x = 0.
26. Which transformation represents y = f(2x + 1)?
a) Shift 1 unit left and stretch horizontally by factor 1/2
b) Shift 1 unit right and stretch horizontally by factor 1/2
c) Shift 1/2 unit left and compress horizontally by factor 2
d) Shift 1/2 unit right and compress horizontally by factor 2
Answer: c) Shift 1/2 unit left and compress horizontally by factor 2
Solution: First, factor out the 2: f(2(x + 1/2)). This shows a horizontal compression by 2 and a shift
left by 1/2.
27. What is the period of y = tan(3x)?
a) π/3
b) π/6
c) 2π/3
d) π
Answer: a) π/3
Solution: The period of tan(kx) is π/k. Here, k = 3, so the period is π/3.
28. Which of the following is neither even nor odd?
a) f(x) = x⁴
b) f(x) = x³
c) f(x) = x² + x
d) f(x) = |x|
Answer: c) f(x) = x² + x
Solution: For this function, f(-x) ≠ f(x) and f(-x) ≠ -f(x), so it's neither even nor odd.
29. What is the x-intercept of f(x) = e^x - 4?
a) ln(4)
b) e⁴
c) 4
d) 0
Answer: a) ln(4)
Solution: Set f(x) = 0 and solve:
e^x - 4 = 0
e^x = 4
x = ln(4)
30. Which function represents a vertical reflection and horizontal stretch by factor 2 of f(x)?
a) -f(x/2)
b) f(-x/2)
c) -f(2x)
d) f(-2x)
Answer: a) -f(x/2)
Solution: -f(x) reflects over x-axis, and f(x/2) stretches horizontally by factor 2.
31. What is the domain of f(x) = arcsin(x)?
a) [-1, 1]
b) (-1, 1)
c) [0, π]
d) (-π/2, π/2)
Answer: a) [-1, 1]
Solution: The arcsine function is defined only for inputs between -1 and 1, inclusive.
32. Which of the following represents an oblique asymptote?
a) lim(x→∞) f(x) = mx + b, where m ≠ 0
b) lim(x→a) f(x) = ∞
c) lim(x→∞) f(x) = b
d) f'(x) = 0
Answer: a) lim(x→∞) f(x) = mx + b, where m ≠ 0
Solution: An oblique asymptote is a slant line that the function approaches as x approaches
infinity.
33. What is the range of f(x) = 3 - e^x?
a) (3, ∞)
b) (-∞, 3)
c) (0, 3)
d) (-∞, ∞)
Answer: b) (-∞, 3)
Solution: As x increases, e^x increases without bound, so 3 - e^x decreases without bound. The
upper limit is 3 (as x approaches -∞).
34. Which transformation represents y = -2f(x - 1) + 3?
a) Reflect over x-axis, stretch vertically by 2, shift right 1, shift up 3
b) Reflect over y-axis, stretch vertically by 2, shift left 1, shift up 3
c) Reflect over x-axis, stretch vertically by 2, shift left 1, shift up 3
d) Reflect over y-axis, compress vertically by 1/2, shift right 1, shift down 3
Answer: c) Reflect over x-axis, stretch vertically by 2, shift left 1, shift up 3
Solution: -2f(x) reflects over x-axis and stretches vertically by 2, (x-1) shifts right by 1, and +3 shifts
up by 3.
8. Which transformation represents y = -f(x)?
a) Reflection over x-axis
b) Reflection over y-axis
c) Vertical stretch by factor of -1
d) Both a and c
Answer: d) Both a and c
Solution: Multiplying f(x) by -1 both reflects the graph over the x-axis and stretches it vertically by a
factor of -1.
9. What is the slope of the line passing through (2, 5) and (4, 9)?
a) 2
b) 4
c) 1
d) 3
Answer: a) 2
Solution: Using the slope formula:
m = (y₂ - y₁) / (x₂ - x₁) = (9 - 5) / (4 - 2) = 4 / 2 = 2
10. Which function represents a horizontal compression by a factor of 3 of f(x)?
a) 3f(x)
b) f(3x)
c) f(x/3)
d) f(x) + 3
Answer: b) f(3x)
Solution: Multiplying x by a constant greater than 1 inside the function compresses the graph
horizontally.
11. What is the vertex of the parabola y = (x - 2)² - 3?
a) (2, -3)
b) (-2, -3)
c) (2, 3)
d) (-2, 3)
Answer: a) (2, -3)
Solution: For y = (x - h)² + k, the vertex is (h, k). Here, h = 2 and k = -3.
12. Which of the following represents an inverse function of f(x) = 2x + 3?
a) f⁻¹(x) = (x - 3)/2
b) f⁻¹(x) = 2x - 3
c) f⁻¹(x) = (x + 3)/2
d) f⁻¹(x) = -2x + 3
Answer: a) f⁻¹(x) = (x - 3)/2
Solution: To find the inverse, swap x and y, then solve for y:
y = 2x + 3
x = 2y + 3
x - 3 = 2y
(x - 3)/2 = y
13. What is the domain of f(x) = 1 / (x² - 4)?
a) All real numbers
b) All real numbers except 2 and -2
c) All real numbers except 0
d) All positive real numbers
Answer: b) All real numbers except 2 and -2
Solution: The denominator cannot be zero:
x² - 4 ≠ 0
(x + 2)(x - 2) ≠ 0
x ≠ 2 and x ≠ -2
14. Which of the following is an odd function?
a) f(x) = x⁴
b) f(x) = x³
c) f(x) = x²
d) f(x) = |x|
Answer: b) f(x) = x³
Solution: For an odd function, f(-x) = -f(x). This is true for x³, as (-x)³ = -x³.
15. What is the range of f(x) = √x?
a) [0, ∞)
b) (-∞, 0]
c) (-∞, ∞)
d) [0, 1]
Answer: a) [0, ∞)
Solution: The square root of a non-negative number is always non-negative, and there's no upper
limit.
16. Which transformation represents y = f(x) + 3?
a) Shift 3 units right
b) Shift 3 units left
c) Shift 3 units up
d) Shift 3 units down
Answer: c) Shift 3 units up
Solution: Adding a constant to the entire function shifts the graph vertically upward by that
amount.
17. What is the period of y = cos(x/2)?
a) π
b) 2π
c) 4π
d) π/2
Answer: c) 4π
Solution: The period of cos(kx) is (2π)/k. Here, k = 1/2, so the period is 2π/(1/2) = 4π.
18. Which function represents a reflection of f(x) over the y-axis?
a) -f(x)
b) f(-x)
c) -f(-x)
d) f(x + 1)
Answer: b) f(-x)
Solution: Negating x inside the function reflects the graph over the y-axis.
19. What is the y-intercept of f(x) = 2x² - 3x + 5?
a) 2
b) -3
c) 5
d) 0
Answer: c) 5
Solution: The y-intercept is f(0):
f(0) = 2(0)² - 3(0) + 5 = 5
20. Which of the following represents a vertical asymptote?
a) lim(x→a) f(x) = ∞
b) lim(x→∞) f(x) = b
c) f(x) = 0
d) f'(x) = 0
Answer: a) lim(x→a) f(x) = ∞
Solution: A vertical asymptote occurs when the function approaches infinity as x approaches a
finite value.
21. What is the slope of the line 3x - 2y = 6?
a) 3/2
b) -3/2
c) 2/3
d) -2/3
Answer: a) 3/2
Solution: Rearrange to slope-intercept form:
-2y = -3x + 6
y = (3/2)x - 3
The coefficient of x is the slope.
22. Which function represents a horizontal reflection and vertical stretch by factor 2 of f(x)?
a) 2f(x)
b) f(2x)
c) 2f(-x)
d) -2f(x)
Answer: c) 2f(-x)
Solution: f(-x) reflects over y-axis, and multiplying by 2 stretches vertically.
23. What is the domain of f(x) = log₂(x - 3)?
a) (3, ∞)
b) [3, ∞)
c) (-∞, 3)
d) (-∞, 3]
Answer: a) (3, ∞)
Solution: The argument of log must be positive:
x - 3 > 0
x > 3
24. Which of the following represents a horizontal asymptote?
a) lim(x→a) f(x) = ∞
b) lim(x→∞) f(x) = b
c) f(x) = 0
d) f'(x) = 0
Answer: b) lim(x→∞) f(x) = b
Solution: A horizontal asymptote occurs when the function approaches a finite value as x
approaches infinity.
25. What is the range of f(x) = 1 / (x² + 1)?
a) (0, 1]
b) [0, 1)
c) (0, ∞)
d) (-∞, ∞)
Answer: a) (0, 1]
Solution: The denominator is always positive and at least 1, so 0 < f(x) ≤ 1, with f(x) = 1 when x = 0.
26. Which transformation represents y = f(2x + 1)?
a) Shift 1 unit left and stretch horizontally by factor 1/2
b) Shift 1 unit right and stretch horizontally by factor 1/2
c) Shift 1/2 unit left and compress horizontally by factor 2
d) Shift 1/2 unit right and compress horizontally by factor 2
Answer: c) Shift 1/2 unit left and compress horizontally by factor 2
Solution: First, factor out the 2: f(2(x + 1/2)). This shows a horizontal compression by 2 and a shift
left by 1/2.
27. What is the period of y = tan(3x)?
a) π/3
b) π/6
c) 2π/3
d) π
Answer: a) π/3
Solution: The period of tan(kx) is π/k. Here, k = 3, so the period is π/3.
28. Which of the following is neither even nor odd?
a) f(x) = x⁴
b) f(x) = x³
c) f(x) = x² + x
d) f(x) = |x|
Answer: c) f(x) = x² + x
Solution: For this function, f(-x) ≠ f(x) and f(-x) ≠ -f(x), so it's neither even nor odd.
29. What is the x-intercept of f(x) = e^x - 4?
a) ln(4)
b) e⁴
c) 4
d) 0
Answer: a) ln(4)
Solution: Set f(x) = 0 and solve:
e^x - 4 = 0
e^x = 4
x = ln(4)
30. Which function represents a vertical reflection and horizontal stretch by factor 2 of f(x)?
a) -f(x/2)
b) f(-x/2)
c) -f(2x)
d) f(-2x)
Answer: a) -f(x/2)
Solution: -f(x) reflects over x-axis, and f(x/2) stretches horizontally by factor 2.
31. What is the domain of f(x) = arcsin(x)?
a) [-1, 1]
b) (-1, 1)
c) [0, π]
d) (-π/2, π/2)
Answer: a) [-1, 1]
Solution: The arcsine function is defined only for inputs between -1 and 1, inclusive.
32. Which of the following represents an oblique asymptote?
a) lim(x→∞) f(x) = mx + b, where m ≠ 0
b) lim(x→a) f(x) = ∞
c) lim(x→∞) f(x) = b
d) f'(x) = 0
Answer: a) lim(x→∞) f(x) = mx + b, where m ≠ 0
Solution: An oblique asymptote is a slant line that the function approaches as x approaches
infinity.
33. What is the range of f(x) = 3 - e^x?
a) (3, ∞)
b) (-∞, 3)
c) (0, 3)
d) (-∞, ∞)
Answer: b) (-∞, 3)
Solution: As x increases, e^x increases without bound, so 3 - e^x decreases without bound. The
upper limit is 3 (as x approaches -∞).
34. Which transformation represents y = -2f(x - 1) + 3?
a) Reflect over x-axis, stretch vertically by 2, shift right 1, shift up 3
b) Reflect over y-axis, stretch vertically by 2, shift left 1, shift up 3
c) Reflect over x-axis, stretch vertically by 2, shift left 1, shift up 3
d) Reflect over y-axis, compress vertically by 1/2, shift right 1, shift down 3
Answer: c) Reflect over x-axis, stretch vertically by 2, shift left 1, shift up 3
Solution: -2f(x) reflects over x-axis and stretches vertically by 2, (x-1) shifts right by 1, and +3 shifts
up by 3.
8. Which transformation represents y = -f(x)?
a) Reflection over x-axis
b) Reflection over y-axis
c) Vertical stretch by factor of -1
d) Both a and c
Answer: d) Both a and c
Solution: Multiplying f(x) by -1 both reflects the graph over the x-axis and stretches it vertically by a
factor of -1.
9. What is the slope of the line passing through (2, 5) and (4, 9)?
a) 2
b) 4
c) 1
d) 3
Answer: a) 2
Solution: Using the slope formula:
m = (y₂ - y₁) / (x₂ - x₁) = (9 - 5) / (4 - 2) = 4 / 2 = 2
10. Which function represents a horizontal compression by a factor of 3 of f(x)?
a) 3f(x)
b) f(3x)
c) f(x/3)
d) f(x) + 3
Answer: b) f(3x)
Solution: Multiplying x by a constant greater than 1 inside the function compresses the graph
horizontally.
11. What is the vertex of the parabola y = (x - 2)² - 3?
a) (2, -3)
b) (-2, -3)
c) (2, 3)
d) (-2, 3)
Answer: a) (2, -3)
Solution: For y = (x - h)² + k, the vertex is (h, k). Here, h = 2 and k = -3.
12. Which of the following represents an inverse function of f(x) = 2x + 3?
a) f⁻¹(x) = (x - 3)/2
b) f⁻¹(x) = 2x - 3
c) f⁻¹(x) = (x + 3)/2
d) f⁻¹(x) = -2x + 3
Answer: a) f⁻¹(x) = (x - 3)/2
Solution: To find the inverse, swap x and y, then solve for y:
y = 2x + 3
x = 2y + 3
x - 3 = 2y
(x - 3)/2 = y
13. What is the domain of f(x) = 1 / (x² - 4)?
a) All real numbers
b) All real numbers except 2 and -2
c) All real numbers except 0
d) All positive real numbers
Answer: b) All real numbers except 2 and -2
Solution: The denominator cannot be zero:
x² - 4 ≠ 0
(x + 2)(x - 2) ≠ 0
x ≠ 2 and x ≠ -2
14. Which of the following is an odd function?
a) f(x) = x⁴
b) f(x) = x³
c) f(x) = x²
d) f(x) = |x|
Answer: b) f(x) = x³
Solution: For an odd function, f(-x) = -f(x). This is true for x³, as (-x)³ = -x³.
15. What is the range of f(x) = √x?
a) [0, ∞)
b) (-∞, 0]
c) (-∞, ∞)
d) [0, 1]
Answer: a) [0, ∞)
Solution: The square root of a non-negative number is always non-negative, and there's no upper
limit.
16. Which transformation represents y = f(x) + 3?
a) Shift 3 units right
b) Shift 3 units left
c) Shift 3 units up
d) Shift 3 units down
Answer: c) Shift 3 units up
Solution: Adding a constant to the entire function shifts the graph vertically upward by that
amount.
17. What is the period of y = cos(x/2)?
a) π
b) 2π
c) 4π
d) π/2
Answer: c) 4π
Solution: The period of cos(kx) is (2π)/k. Here, k = 1/2, so the period is 2π/(1/2) = 4π.
18. Which function represents a reflection of f(x) over the y-axis?
a) -f(x)
b) f(-x)
c) -f(-x)
d) f(x + 1)
Answer: b) f(-x)
Solution: Negating x inside the function reflects the graph over the y-axis.
19. What is the y-intercept of f(x) = 2x² - 3x + 5?
a) 2
b) -3
c) 5
d) 0
Answer: c) 5
Solution: The y-intercept is f(0):
f(0) = 2(0)² - 3(0) + 5 = 5
20. Which of the following represents a vertical asymptote?
a) lim(x→a) f(x) = ∞
b) lim(x→∞) f(x) = b
c) f(x) = 0
d) f'(x) = 0
Answer: a) lim(x→a) f(x) = ∞
Solution: A vertical asymptote occurs when the function approaches infinity as x approaches a
finite value.
21. What is the slope of the line 3x - 2y = 6?
a) 3/2
b) -3/2
c) 2/3
d) -2/3
Answer: a) 3/2
Solution: Rearrange to slope-intercept form:
-2y = -3x + 6
y = (3/2)x - 3
The coefficient of x is the slope.
22. Which function represents a horizontal reflection and vertical stretch by factor 2 of f(x)?
a) 2f(x)
b) f(2x)
c) 2f(-x)
d) -2f(x)
Answer: c) 2f(-x)
Solution: f(-x) reflects over y-axis, and multiplying by 2 stretches vertically.
23. What is the domain of f(x) = log₂(x - 3)?
a) (3, ∞)
b) [3, ∞)
c) (-∞, 3)
d) (-∞, 3]
Answer: a) (3, ∞)
Solution: The argument of log must be positive:
x - 3 > 0
x > 3
24. Which of the following represents a horizontal asymptote?
a) lim(x→a) f(x) = ∞
b) lim(x→∞) f(x) = b
c) f(x) = 0
d) f'(x) = 0
Answer: b) lim(x→∞) f(x) = b
Solution: A horizontal asymptote occurs when the function approaches a finite value as x
approaches infinity.
25. What is the range of f(x) = 1 / (x² + 1)?
a) (0, 1]
b) [0, 1)
c) (0, ∞)
d) (-∞, ∞)
Answer: a) (0, 1]
Solution: The denominator is always positive and at least 1, so 0 < f(x) ≤ 1, with f(x) = 1 when x = 0.
26. Which transformation represents y = f(2x + 1)?
a) Shift 1 unit left and stretch horizontally by factor 1/2
b) Shift 1 unit right and stretch horizontally by factor 1/2
c) Shift 1/2 unit left and compress horizontally by factor 2
d) Shift 1/2 unit right and compress horizontally by factor 2
Answer: c) Shift 1/2 unit left and compress horizontally by factor 2
Solution: First, factor out the 2: f(2(x + 1/2)). This shows a horizontal compression by 2 and a shift
left by 1/2.
27. What is the period of y = tan(3x)?
a) π/3
b) π/6
c) 2π/3
d) π
Answer: a) π/3
Solution: The period of tan(kx) is π/k. Here, k = 3, so the period is π/3.
28. Which of the following is neither even nor odd?
a) f(x) = x⁴
b) f(x) = x³
c) f(x) = x² + x
d) f(x) = |x|
Answer: c) f(x) = x² + x
Solution: For this function, f(-x) ≠ f(x) and f(-x) ≠ -f(x), so it's neither even nor odd.
29. What is the x-intercept of f(x) = e^x - 4?
a) ln(4)
b) e⁴
c) 4
d) 0
Answer: a) ln(4)
Solution: Set f(x) = 0 and solve:
e^x - 4 = 0
e^x = 4
x = ln(4)
30. Which function represents a vertical reflection and horizontal stretch by factor 2 of f(x)?
a) -f(x/2)
b) f(-x/2)
c) -f(2x)
d) f(-2x)
Answer: a) -f(x/2)
Solution: -f(x) reflects over x-axis, and f(x/2) stretches horizontally by factor 2.
31. What is the domain of f(x) = arcsin(x)?
a) [-1, 1]
b) (-1, 1)
c) [0, π]
d) (-π/2, π/2)
Answer: a) [-1, 1]
Solution: The arcsine function is defined only for inputs between -1 and 1, inclusive.
32. Which of the following represents an oblique asymptote?
a) lim(x→∞) f(x) = mx + b, where m ≠ 0
b) lim(x→a) f(x) = ∞
c) lim(x→∞) f(x) = b
d) f'(x) = 0
Answer: a) lim(x→∞) f(x) = mx + b, where m ≠ 0
Solution: An oblique asymptote is a slant line that the function approaches as x approaches
infinity.
33. What is the range of f(x) = 3 - e^x?
a) (3, ∞)
b) (-∞, 3)
c) (0, 3)
d) (-∞, ∞)
Answer: b) (-∞, 3)
Solution: As x increases, e^x increases without bound, so 3 - e^x decreases without bound. The
upper limit is 3 (as x approaches -∞).
34. Which transformation represents y = -2f(x - 1) + 3?
a) Reflect over x-axis, stretch vertically by 2, shift right 1, shift up 3
b) Reflect over y-axis, stretch vertically by 2, shift left 1, shift up 3
c) Reflect over x-axis, stretch vertically by 2, shift left 1, shift up 3
d) Reflect over y-axis, compress vertically by 1/2, shift right 1, shift down 3
Answer: c) Reflect over x-axis, stretch vertically by 2, shift left 1, shift up 3
Solution: -2f(x) reflects over x-axis and stretches vertically by 2, (x-1) shifts right by 1, and +3 shifts
up by 3.
8. Which transformation represents y = -f(x)?
a) Reflection over x-axis
b) Reflection over y-axis
c) Vertical stretch by factor of -1
d) Both a and c
Answer: d) Both a and c
Solution: Multiplying f(x) by -1 both reflects the graph over the x-axis and stretches it vertically by a
factor of -1.
9. What is the slope of the line passing through (2, 5) and (4, 9)?
a) 2
b) 4
c) 1
d) 3
Answer: a) 2
Solution: Using the slope formula:
m = (y₂ - y₁) / (x₂ - x₁) = (9 - 5) / (4 - 2) = 4 / 2 = 2
10. Which function represents a horizontal compression by a factor of 3 of f(x)?
a) 3f(x)
b) f(3x)
c) f(x/3)
d) f(x) + 3
Answer: b) f(3x)
Solution: Multiplying x by a constant greater than 1 inside the function compresses the graph
horizontally.
11. What is the vertex of the parabola y = (x - 2)² - 3?
a) (2, -3)
b) (-2, -3)
c) (2, 3)
d) (-2, 3)
Answer: a) (2, -3)
Solution: For y = (x - h)² + k, the vertex is (h, k). Here, h = 2 and k = -3.
12. Which of the following represents an inverse function of f(x) = 2x + 3?
a) f⁻¹(x) = (x - 3)/2
b) f⁻¹(x) = 2x - 3
c) f⁻¹(x) = (x + 3)/2
d) f⁻¹(x) = -2x + 3
Answer: a) f⁻¹(x) = (x - 3)/2
Solution: To find the inverse, swap x and y, then solve for y:
y = 2x + 3
x = 2y + 3
x - 3 = 2y
(x - 3)/2 = y
13. What is the domain of f(x) = 1 / (x² - 4)?
a) All real numbers
b) All real numbers except 2 and -2
c) All real numbers except 0
d) All positive real numbers
Answer: b) All real numbers except 2 and -2
Solution: The denominator cannot be zero:
x² - 4 ≠ 0
(x + 2)(x - 2) ≠ 0
x ≠ 2 and x ≠ -2
14. Which of the following is an odd function?
a) f(x) = x⁴
b) f(x) = x³
c) f(x) = x²
d) f(x) = |x|
Answer: b) f(x) = x³
Solution: For an odd function, f(-x) = -f(x). This is true for x³, as (-x)³ = -x³.
15. What is the range of f(x) = √x?
a) [0, ∞)
b) (-∞, 0]
c) (-∞, ∞)
d) [0, 1]
Answer: a) [0, ∞)
Solution: The square root of a non-negative number is always non-negative, and there's no upper
limit.
16. Which transformation represents y = f(x) + 3?
a) Shift 3 units right
b) Shift 3 units left
c) Shift 3 units up
d) Shift 3 units down
Answer: c) Shift 3 units up
Solution: Adding a constant to the entire function shifts the graph vertically upward by that
amount.
17. What is the period of y = cos(x/2)?
a) π
b) 2π
c) 4π
d) π/2
Answer: c) 4π
Solution: The period of cos(kx) is (2π)/k. Here, k = 1/2, so the period is 2π/(1/2) = 4π.
18. Which function represents a reflection of f(x) over the y-axis?
a) -f(x)
b) f(-x)
c) -f(-x)
d) f(x + 1)
Answer: b) f(-x)
Solution: Negating x inside the function reflects the graph over the y-axis.
19. What is the y-intercept of f(x) = 2x² - 3x + 5?
a) 2
b) -3
c) 5
d) 0
Answer: c) 5
Solution: The y-intercept is f(0):
f(0) = 2(0)² - 3(0) + 5 = 5
20. Which of the following represents a vertical asymptote?
a) lim(x→a) f(x) = ∞
b) lim(x→∞) f(x) = b
c) f(x) = 0
d) f'(x) = 0
Answer: a) lim(x→a) f(x) = ∞
Solution: A vertical asymptote occurs when the function approaches infinity as x approaches a
finite value.
21. What is the slope of the line 3x - 2y = 6?
a) 3/2
b) -3/2
c) 2/3
d) -2/3
Answer: a) 3/2
Solution: Rearrange to slope-intercept form:
-2y = -3x + 6
y = (3/2)x - 3
The coefficient of x is the slope.
22. Which function represents a horizontal reflection and vertical stretch by factor 2 of f(x)?
a) 2f(x)
b) f(2x)
c) 2f(-x)
d) -2f(x)
Answer: c) 2f(-x)
Solution: f(-x) reflects over y-axis, and multiplying by 2 stretches vertically.
23. What is the domain of f(x) = log₂(x - 3)?
a) (3, ∞)
b) [3, ∞)
c) (-∞, 3)
d) (-∞, 3]
Answer: a) (3, ∞)
Solution: The argument of log must be positive:
x - 3 > 0
x > 3
24. Which of the following represents a horizontal asymptote?
a) lim(x→a) f(x) = ∞
b) lim(x→∞) f(x) = b
c) f(x) = 0
d) f'(x) = 0
Answer: b) lim(x→∞) f(x) = b
Solution: A horizontal asymptote occurs when the function approaches a finite value as x
approaches infinity.
25. What is the range of f(x) = 1 / (x² + 1)?
a) (0, 1]
b) [0, 1)
c) (0, ∞)
d) (-∞, ∞)
Answer: a) (0, 1]
Solution: The denominator is always positive and at least 1, so 0 < f(x) ≤ 1, with f(x) = 1 when x = 0.
26. Which transformation represents y = f(2x + 1)?
a) Shift 1 unit left and stretch horizontally by factor 1/2
b) Shift 1 unit right and stretch horizontally by factor 1/2
c) Shift 1/2 unit left and compress horizontally by factor 2
d) Shift 1/2 unit right and compress horizontally by factor 2
Answer: c) Shift 1/2 unit left and compress horizontally by factor 2
Solution: First, factor out the 2: f(2(x + 1/2)). This shows a horizontal compression by 2 and a shift
left by 1/2.
27. What is the period of y = tan(3x)?
a) π/3
b) π/6
c) 2π/3
d) π
Answer: a) π/3
Solution: The period of tan(kx) is π/k. Here, k = 3, so the period is π/3.
28. Which of the following is neither even nor odd?
a) f(x) = x⁴
b) f(x) = x³
c) f(x) = x² + x
d) f(x) = |x|
Answer: c) f(x) = x² + x
Solution: For this function, f(-x) ≠ f(x) and f(-x) ≠ -f(x), so it's neither even nor odd.
29. What is the x-intercept of f(x) = e^x - 4?
a) ln(4)
b) e⁴
c) 4
d) 0
Answer: a) ln(4)
Solution: Set f(x) = 0 and solve:
e^x - 4 = 0
e^x = 4
x = ln(4)
30. Which function represents a vertical reflection and horizontal stretch by factor 2 of f(x)?
a) -f(x/2)
b) f(-x/2)
c) -f(2x)
d) f(-2x)
Answer: a) -f(x/2)
Solution: -f(x) reflects over x-axis, and f(x/2) stretches horizontally by factor 2.
31. What is the domain of f(x) = arcsin(x)?
a) [-1, 1]
b) (-1, 1)
c) [0, π]
d) (-π/2, π/2)
Answer: a) [-1, 1]
Solution: The arcsine function is defined only for inputs between -1 and 1, inclusive.
32. Which of the following represents an oblique asymptote?
a) lim(x→∞) f(x) = mx + b, where m ≠ 0
b) lim(x→a) f(x) = ∞
c) lim(x→∞) f(x) = b
d) f'(x) = 0
Answer: a) lim(x→∞) f(x) = mx + b, where m ≠ 0
Solution: An oblique asymptote is a slant line that the function approaches as x approaches
infinity.
33. What is the range of f(x) = 3 - e^x?
a) (3, ∞)
b) (-∞, 3)
c) (0, 3)
d) (-∞, ∞)
Answer: b) (-∞, 3)
Solution: As x increases, e^x increases without bound, so 3 - e^x decreases without bound. The
upper limit is 3 (as x approaches -∞).
34. Which transformation represents y = -2f(x - 1) + 3?
a) Reflect over x-axis, stretch vertically by 2, shift right 1, shift up 3
b) Reflect over y-axis, stretch vertically by 2, shift left 1, shift up 3
c) Reflect over x-axis, stretch vertically by 2, shift left 1, shift up 3
d) Reflect over y-axis, compress vertically by 1/2, shift right 1, shift down 3
Answer: c) Reflect over x-axis, stretch vertically by 2, shift left 1, shift up 3
Solution: -2f(x) reflects over x-axis and stretches vertically by 2, (x-1) shifts right by 1, and +3 shifts
up by 3.
8. Which transformation represents y = -f(x)?
a) Reflection over x-axis
b) Reflection over y-axis
c) Vertical stretch by factor of -1
d) Both a and c
Answer: d) Both a and c
Solution: Multiplying f(x) by -1 both reflects the graph over the x-axis and stretches it vertically by a
factor of -1.
9. What is the slope of the line passing through (2, 5) and (4, 9)?
a) 2
b) 4
c) 1
d) 3
Answer: a) 2
Solution: Using the slope formula:
m = (y₂ - y₁) / (x₂ - x₁) = (9 - 5) / (4 - 2) = 4 / 2 = 2
10. Which function represents a horizontal compression by a factor of 3 of f(x)?
a) 3f(x)
b) f(3x)
c) f(x/3)
d) f(x) + 3
Answer: b) f(3x)
Solution: Multiplying x by a constant greater than 1 inside the function compresses the graph
horizontally.
11. What is the vertex of the parabola y = (x - 2)² - 3?
a) (2, -3)
b) (-2, -3)
c) (2, 3)
d) (-2, 3)
Answer: a) (2, -3)
Solution: For y = (x - h)² + k, the vertex is (h, k). Here, h = 2 and k = -3.
12. Which of the following represents an inverse function of f(x) = 2x + 3?
a) f⁻¹(x) = (x - 3)/2
b) f⁻¹(x) = 2x - 3
c) f⁻¹(x) = (x + 3)/2
d) f⁻¹(x) = -2x + 3
Answer: a) f⁻¹(x) = (x - 3)/2
Solution: To find the inverse, swap x and y, then solve for y:
y = 2x + 3
x = 2y + 3
x - 3 = 2y
(x - 3)/2 = y
13. What is the domain of f(x) = 1 / (x² - 4)?
a) All real numbers
b) All real numbers except 2 and -2
c) All real numbers except 0
d) All positive real numbers
Answer: b) All real numbers except 2 and -2
Solution: The denominator cannot be zero:
x² - 4 ≠ 0
(x + 2)(x - 2) ≠ 0
x ≠ 2 and x ≠ -2
14. Which of the following is an odd function?
a) f(x) = x⁴
b) f(x) = x³
c) f(x) = x²
d) f(x) = |x|
Answer: b) f(x) = x³
Solution: For an odd function, f(-x) = -f(x). This is true for x³, as (-x)³ = -x³.
15. What is the range of f(x) = √x?
a) [0, ∞)
b) (-∞, 0]
c) (-∞, ∞)
d) [0, 1]
Answer: a) [0, ∞)
Solution: The square root of a non-negative number is always non-negative, and there's no upper
limit.
16. Which transformation represents y = f(x) + 3?
a) Shift 3 units right
b) Shift 3 units left
c) Shift 3 units up
d) Shift 3 units down
Answer: c) Shift 3 units up
Solution: Adding a constant to the entire function shifts the graph vertically upward by that
amount.
17. What is the period of y = cos(x/2)?
a) π
b) 2π
c) 4π
d) π/2
Answer: c) 4π
Solution: The period of cos(kx) is (2π)/k. Here, k = 1/2, so the period is 2π/(1/2) = 4π.
18. Which function represents a reflection of f(x) over the y-axis?
a) -f(x)
b) f(-x)
c) -f(-x)
d) f(x + 1)
Answer: b) f(-x)
Solution: Negating x inside the function reflects the graph over the y-axis.
19. What is the y-intercept of f(x) = 2x² - 3x + 5?
a) 2
b) -3
c) 5
d) 0
Answer: c) 5
Solution: The y-intercept is f(0):
f(0) = 2(0)² - 3(0) + 5 = 5
20. Which of the following represents a vertical asymptote?
a) lim(x→a) f(x) = ∞
b) lim(x→∞) f(x) = b
c) f(x) = 0
d) f'(x) = 0
Answer: a) lim(x→a) f(x) = ∞
Solution: A vertical asymptote occurs when the function approaches infinity as x approaches a
finite value.
21. What is the slope of the line 3x - 2y = 6?
a) 3/2
b) -3/2
c) 2/3
d) -2/3
Answer: a) 3/2
Solution: Rearrange to slope-intercept form:
-2y = -3x + 6
y = (3/2)x - 3
The coefficient of x is the slope.
22. Which function represents a horizontal reflection and vertical stretch by factor 2 of f(x)?
a) 2f(x)
b) f(2x)
c) 2f(-x)
d) -2f(x)
Answer: c) 2f(-x)
Solution: f(-x) reflects over y-axis, and multiplying by 2 stretches vertically.
23. What is the domain of f(x) = log₂(x - 3)?
a) (3, ∞)
b) [3, ∞)
c) (-∞, 3)
d) (-∞, 3]
Answer: a) (3, ∞)
Solution: The argument of log must be positive:
x - 3 > 0
x > 3
24. Which of the following represents a horizontal asymptote?
a) lim(x→a) f(x) = ∞
b) lim(x→∞) f(x) = b
c) f(x) = 0
d) f'(x) = 0
Answer: b) lim(x→∞) f(x) = b
Solution: A horizontal asymptote occurs when the function approaches a finite value as x
approaches infinity.
25. What is the range of f(x) = 1 / (x² + 1)?
a) (0, 1]
b) [0, 1)
c) (0, ∞)
d) (-∞, ∞)
Answer: a) (0, 1]
Solution: The denominator is always positive and at least 1, so 0 < f(x) ≤ 1, with f(x) = 1 when x = 0.
26. Which transformation represents y = f(2x + 1)?
a) Shift 1 unit left and stretch horizontally by factor 1/2
b) Shift 1 unit right and stretch horizontally by factor 1/2
c) Shift 1/2 unit left and compress horizontally by factor 2
d) Shift 1/2 unit right and compress horizontally by factor 2
Answer: c) Shift 1/2 unit left and compress horizontally by factor 2
Solution: First, factor out the 2: f(2(x + 1/2)). This shows a horizontal compression by 2 and a shift
left by 1/2.
27. What is the period of y = tan(3x)?
a) π/3
b) π/6
c) 2π/3
d) π
Answer: a) π/3
Solution: The period of tan(kx) is π/k. Here, k = 3, so the period is π/3.
28. Which of the following is neither even nor odd?
a) f(x) = x⁴
b) f(x) = x³
c) f(x) = x² + x
d) f(x) = |x|
Answer: c) f(x) = x² + x
Solution: For this function, f(-x) ≠ f(x) and f(-x) ≠ -f(x), so it's neither even nor odd.
29. What is the x-intercept of f(x) = e^x - 4?
a) ln(4)
b) e⁴
c) 4
d) 0
Answer: a) ln(4)
Solution: Set f(x) = 0 and solve:
e^x - 4 = 0
e^x = 4
x = ln(4)
30. Which function represents a vertical reflection and horizontal stretch by factor 2 of f(x)?
a) -f(x/2)
b) f(-x/2)
c) -f(2x)
d) f(-2x)
Answer: a) -f(x/2)
Solution: -f(x) reflects over x-axis, and f(x/2) stretches horizontally by factor 2.
31. What is the domain of f(x) = arcsin(x)?
a) [-1, 1]
b) (-1, 1)
c) [0, π]
d) (-π/2, π/2)
Answer: a) [-1, 1]
Solution: The arcsine function is defined only for inputs between -1 and 1, inclusive.
32. Which of the following represents an oblique asymptote?
a) lim(x→∞) f(x) = mx + b, where m ≠ 0
b) lim(x→a) f(x) = ∞
c) lim(x→∞) f(x) = b
d) f'(x) = 0
Answer: a) lim(x→∞) f(x) = mx + b, where m ≠ 0
Solution: An oblique asymptote is a slant line that the function approaches as x approaches
infinity.
33. What is the range of f(x) = 3 - e^x?
a) (3, ∞)
b) (-∞, 3)
c) (0, 3)
d) (-∞, ∞)
Answer: b) (-∞, 3)
Solution: As x increases, e^x increases without bound, so 3 - e^x decreases without bound. The
upper limit is 3 (as x approaches -∞).
34. Which transformation represents y = -2f(x - 1) + 3?
a) Reflect over x-axis, stretch vertically by 2, shift right 1, shift up 3
b) Reflect over y-axis, stretch vertically by 2, shift left 1, shift up 3
c) Reflect over x-axis, stretch vertically by 2, shift left 1, shift up 3
d) Reflect over y-axis, compress vertically by 1/2, shift right 1, shift down 3
Answer: c) Reflect over x-axis, stretch vertically by 2, shift left 1, shift up 3
Solution: -2f(x) reflects over x-axis and stretches vertically by 2, (x-1) shifts right by 1, and +3 shifts
up by 3.
8. Which transformation represents y = -f(x)?
a) Reflection over x-axis
b) Reflection over y-axis
c) Vertical stretch by factor of -1
d) Both a and c
Answer: d) Both a and c
Solution: Multiplying f(x) by -1 both reflects the graph over the x-axis and stretches it vertically by a
factor of -1.
9. What is the slope of the line passing through (2, 5) and (4, 9)?
a) 2
b) 4
c) 1
d) 3
Answer: a) 2
Solution: Using the slope formula:
m = (y₂ - y₁) / (x₂ - x₁) = (9 - 5) / (4 - 2) = 4 / 2 = 2
10. Which function represents a horizontal compression by a factor of 3 of f(x)?
a) 3f(x)
b) f(3x)
c) f(x/3)
d) f(x) + 3
Answer: b) f(3x)
Solution: Multiplying x by a constant greater than 1 inside the function compresses the graph
horizontally.
11. What is the vertex of the parabola y = (x - 2)² - 3?
a) (2, -3)
b) (-2, -3)
c) (2, 3)
d) (-2, 3)
Answer: a) (2, -3)
Solution: For y = (x - h)² + k, the vertex is (h, k). Here, h = 2 and k = -3.
12. Which of the following represents an inverse function of f(x) = 2x + 3?
a) f⁻¹(x) = (x - 3)/2
b) f⁻¹(x) = 2x - 3
c) f⁻¹(x) = (x + 3)/2
d) f⁻¹(x) = -2x + 3
Answer: a) f⁻¹(x) = (x - 3)/2
Solution: To find the inverse, swap x and y, then solve for y:
y = 2x + 3
x = 2y + 3
x - 3 = 2y
(x - 3)/2 = y
13. What is the domain of f(x) = 1 / (x² - 4)?
a) All real numbers
b) All real numbers except 2 and -2
c) All real numbers except 0
d) All positive real numbers
Answer: b) All real numbers except 2 and -2
Solution: The denominator cannot be zero:
x² - 4 ≠ 0
(x + 2)(x - 2) ≠ 0
x ≠ 2 and x ≠ -2
14. Which of the following is an odd function?
a) f(x) = x⁴
b) f(x) = x³
c) f(x) = x²
d) f(x) = |x|
Answer: b) f(x) = x³
Solution: For an odd function, f(-x) = -f(x). This is true for x³, as (-x)³ = -x³.
15. What is the range of f(x) = √x?
a) [0, ∞)
b) (-∞, 0]
c) (-∞, ∞)
d) [0, 1]
Answer: a) [0, ∞)
Solution: The square root of a non-negative number is always non-negative, and there's no upper
limit.
16. Which transformation represents y = f(x) + 3?
a) Shift 3 units right
b) Shift 3 units left
c) Shift 3 units up
d) Shift 3 units down
Answer: c) Shift 3 units up
Solution: Adding a constant to the entire function shifts the graph vertically upward by that
amount.
17. What is the period of y = cos(x/2)?
a) π
b) 2π
c) 4π
d) π/2
Answer: c) 4π
Solution: The period of cos(kx) is (2π)/k. Here, k = 1/2, so the period is 2π/(1/2) = 4π.
18. Which function represents a reflection of f(x) over the y-axis?
a) -f(x)
b) f(-x)
c) -f(-x)
d) f(x + 1)
Answer: b) f(-x)
Solution: Negating x inside the function reflects the graph over the y-axis.
19. What is the y-intercept of f(x) = 2x² - 3x + 5?
a) 2
b) -3
c) 5
d) 0
Answer: c) 5
Solution: The y-intercept is f(0):
f(0) = 2(0)² - 3(0) + 5 = 5
20. Which of the following represents a vertical asymptote?
a) lim(x→a) f(x) = ∞
b) lim(x→∞) f(x) = b
c) f(x) = 0
d) f'(x) = 0
Answer: a) lim(x→a) f(x) = ∞
Solution: A vertical asymptote occurs when the function approaches infinity as x approaches a
finite value.
21. What is the slope of the line 3x - 2y = 6?
a) 3/2
b) -3/2
c) 2/3
d) -2/3
Answer: a) 3/2
Solution: Rearrange to slope-intercept form:
-2y = -3x + 6
y = (3/2)x - 3
The coefficient of x is the slope.
22. Which function represents a horizontal reflection and vertical stretch by factor 2 of f(x)?
a) 2f(x)
b) f(2x)
c) 2f(-x)
d) -2f(x)
Answer: c) 2f(-x)
Solution: f(-x) reflects over y-axis, and multiplying by 2 stretches vertically.
23. What is the domain of f(x) = log₂(x - 3)?
a) (3, ∞)
b) [3, ∞)
c) (-∞, 3)
d) (-∞, 3]
Answer: a) (3, ∞)
Solution: The argument of log must be positive:
x - 3 > 0
x > 3
24. Which of the following represents a horizontal asymptote?
a) lim(x→a) f(x) = ∞
b) lim(x→∞) f(x) = b
c) f(x) = 0
d) f'(x) = 0
Answer: b) lim(x→∞) f(x) = b
Solution: A horizontal asymptote occurs when the function approaches a finite value as x
approaches infinity.
25. What is the range of f(x) = 1 / (x² + 1)?
a) (0, 1]
b) [0, 1)
c) (0, ∞)
d) (-∞, ∞)
Answer: a) (0, 1]
Solution: The denominator is always positive and at least 1, so 0 < f(x) ≤ 1, with f(x) = 1 when x = 0.
26. Which transformation represents y = f(2x + 1)?
a) Shift 1 unit left and stretch horizontally by factor 1/2
b) Shift 1 unit right and stretch horizontally by factor 1/2
c) Shift 1/2 unit left and compress horizontally by factor 2
d) Shift 1/2 unit right and compress horizontally by factor 2
Answer: c) Shift 1/2 unit left and compress horizontally by factor 2
Solution: First, factor out the 2: f(2(x + 1/2)). This shows a horizontal compression by 2 and a shift
left by 1/2.
27. What is the period of y = tan(3x)?
a) π/3
b) π/6
c) 2π/3
d) π
Answer: a) π/3
Solution: The period of tan(kx) is π/k. Here, k = 3, so the period is π/3.
28. Which of the following is neither even nor odd?
a) f(x) = x⁴
b) f(x) = x³
c) f(x) = x² + x
d) f(x) = |x|
Answer: c) f(x) = x² + x
Solution: For this function, f(-x) ≠ f(x) and f(-x) ≠ -f(x), so it's neither even nor odd.
29. What is the x-intercept of f(x) = e^x - 4?
a) ln(4)
b) e⁴
c) 4
d) 0
Answer: a) ln(4)
Solution: Set f(x) = 0 and solve:
e^x - 4 = 0
e^x = 4
x = ln(4)
30. Which function represents a vertical reflection and horizontal stretch by factor 2 of f(x)?
a) -f(x/2)
b) f(-x/2)
c) -f(2x)
d) f(-2x)
Answer: a) -f(x/2)
Solution: -f(x) reflects over x-axis, and f(x/2) stretches horizontally by factor 2.
31. What is the domain of f(x) = arcsin(x)?
a) [-1, 1]
b) (-1, 1)
c) [0, π]
d) (-π/2, π/2)
Answer: a) [-1, 1]
Solution: The arcsine function is defined only for inputs between -1 and 1, inclusive.
32. Which of the following represents an oblique asymptote?
a) lim(x→∞) f(x) = mx + b, where m ≠ 0
b) lim(x→a) f(x) = ∞
c) lim(x→∞) f(x) = b
d) f'(x) = 0
Answer: a) lim(x→∞) f(x) = mx + b, where m ≠ 0
Solution: An oblique asymptote is a slant line that the function approaches as x approaches
infinity.
33. What is the range of f(x) = 3 - e^x?
a) (3, ∞)
b) (-∞, 3)
c) (0, 3)
d) (-∞, ∞)
Answer: b) (-∞, 3)
Solution: As x increases, e^x increases without bound, so 3 - e^x decreases without bound. The
upper limit is 3 (as x approaches -∞).
34. Which transformation represents y = -2f(x - 1) + 3?
a) Reflect over x-axis, stretch vertically by 2, shift right 1, shift up 3
b) Reflect over y-axis, stretch vertically by 2, shift left 1, shift up 3
c) Reflect over x-axis, stretch vertically by 2, shift left 1, shift up 3
d) Reflect over y-axis, compress vertically by 1/2, shift right 1, shift down 3
Answer: c) Reflect over x-axis, stretch vertically by 2, shift left 1, shift up 3
Solution: -2f(x) reflects over x-axis and stretches vertically by 2, (x-1) shifts right by 1, and +3 shifts
up by 3.
8. Which transformation represents y = -f(x)?
a) Reflection over x-axis
b) Reflection over y-axis
c) Vertical stretch by factor of -1
d) Both a and c
Answer: d) Both a and c
Solution: Multiplying f(x) by -1 both reflects the graph over the x-axis and stretches it vertically by a
factor of -1.
9. What is the slope of the line passing through (2, 5) and (4, 9)?
a) 2
b) 4
c) 1
d) 3
Answer: a) 2
Solution: Using the slope formula:
m = (y₂ - y₁) / (x₂ - x₁) = (9 - 5) / (4 - 2) = 4 / 2 = 2
10. Which function represents a horizontal compression by a factor of 3 of f(x)?
a) 3f(x)
b) f(3x)
c) f(x/3)
d) f(x) + 3
Answer: b) f(3x)
Solution: Multiplying x by a constant greater than 1 inside the function compresses the graph
horizontally.
11. What is the vertex of the parabola y = (x - 2)² - 3?
a) (2, -3)
b) (-2, -3)
c) (2, 3)
d) (-2, 3)
Answer: a) (2, -3)
Solution: For y = (x - h)² + k, the vertex is (h, k). Here, h = 2 and k = -3.
12. Which of the following represents an inverse function of f(x) = 2x + 3?
a) f⁻¹(x) = (x - 3)/2
b) f⁻¹(x) = 2x - 3
c) f⁻¹(x) = (x + 3)/2
d) f⁻¹(x) = -2x + 3
Answer: a) f⁻¹(x) = (x - 3)/2
Solution: To find the inverse, swap x and y, then solve for y:
y = 2x + 3
x = 2y + 3
x - 3 = 2y
(x - 3)/2 = y
13. What is the domain of f(x) = 1 / (x² - 4)?
a) All real numbers
b) All real numbers except 2 and -2
c) All real numbers except 0
d) All positive real numbers
Answer: b) All real numbers except 2 and -2
Solution: The denominator cannot be zero:
x² - 4 ≠ 0
(x + 2)(x - 2) ≠ 0
x ≠ 2 and x ≠ -2
14. Which of the following is an odd function?
a) f(x) = x⁴
b) f(x) = x³
c) f(x) = x²
d) f(x) = |x|
Answer: b) f(x) = x³
Solution: For an odd function, f(-x) = -f(x). This is true for x³, as (-x)³ = -x³.
15. What is the range of f(x) = √x?
a) [0, ∞)
b) (-∞, 0]
c) (-∞, ∞)
d) [0, 1]
Answer: a) [0, ∞)
Solution: The square root of a non-negative number is always non-negative, and there's no upper
limit.
16. Which transformation represents y = f(x) + 3?
a) Shift 3 units right
b) Shift 3 units left
c) Shift 3 units up
d) Shift 3 units down
Answer: c) Shift 3 units up
Solution: Adding a constant to the entire function shifts the graph vertically upward by that
amount.
17. What is the period of y = cos(x/2)?
a) π
b) 2π
c) 4π
d) π/2
Answer: c) 4π
Solution: The period of cos(kx) is (2π)/k. Here, k = 1/2, so the period is 2π/(1/2) = 4π.
18. Which function represents a reflection of f(x) over the y-axis?
a) -f(x)
b) f(-x)
c) -f(-x)
d) f(x + 1)
Answer: b) f(-x)
Solution: Negating x inside the function reflects the graph over the y-axis.
19. What is the y-intercept of f(x) = 2x² - 3x + 5?
a) 2
b) -3
c) 5
d) 0
Answer: c) 5
Solution: The y-intercept is f(0):
f(0) = 2(0)² - 3(0) + 5 = 5
20. Which of the following represents a vertical asymptote?
a) lim(x→a) f(x) = ∞
b) lim(x→∞) f(x) = b
c) f(x) = 0
d) f'(x) = 0
Answer: a) lim(x→a) f(x) = ∞
Solution: A vertical asymptote occurs when the function approaches infinity as x approaches a
finite value.
21. What is the slope of the line 3x - 2y = 6?
a) 3/2
b) -3/2
c) 2/3
d) -2/3
Answer: a) 3/2
Solution: Rearrange to slope-intercept form:
-2y = -3x + 6
y = (3/2)x - 3
The coefficient of x is the slope.
22. Which function represents a horizontal reflection and vertical stretch by factor 2 of f(x)?
a) 2f(x)
b) f(2x)
c) 2f(-x)
d) -2f(x)
Answer: c) 2f(-x)
Solution: f(-x) reflects over y-axis, and multiplying by 2 stretches vertically.
23. What is the domain of f(x) = log₂(x - 3)?
a) (3, ∞)
b) [3, ∞)
c) (-∞, 3)
d) (-∞, 3]
Answer: a) (3, ∞)
Solution: The argument of log must be positive:
x - 3 > 0
x > 3
24. Which of the following represents a horizontal asymptote?
a) lim(x→a) f(x) = ∞
b) lim(x→∞) f(x) = b
c) f(x) = 0
d) f'(x) = 0
Answer: b) lim(x→∞) f(x) = b
Solution: A horizontal asymptote occurs when the function approaches a finite value as x
approaches infinity.
25. What is the range of f(x) = 1 / (x² + 1)?
a) (0, 1]
b) [0, 1)
c) (0, ∞)
d) (-∞, ∞)
Answer: a) (0, 1]
Solution: The denominator is always positive and at least 1, so 0 < f(x) ≤ 1, with f(x) = 1 when x = 0.
26. Which transformation represents y = f(2x + 1)?
a) Shift 1 unit left and stretch horizontally by factor 1/2
b) Shift 1 unit right and stretch horizontally by factor 1/2
c) Shift 1/2 unit left and compress horizontally by factor 2
d) Shift 1/2 unit right and compress horizontally by factor 2
Answer: c) Shift 1/2 unit left and compress horizontally by factor 2
Solution: First, factor out the 2: f(2(x + 1/2)). This shows a horizontal compression by 2 and a shift
left by 1/2.
27. What is the period of y = tan(3x)?
a) π/3
b) π/6
c) 2π/3
d) π
Answer: a) π/3
Solution: The period of tan(kx) is π/k. Here, k = 3, so the period is π/3.
28. Which of the following is neither even nor odd?
a) f(x) = x⁴
b) f(x) = x³
c) f(x) = x² + x
d) f(x) = |x|
Answer: c) f(x) = x² + x
Solution: For this function, f(-x) ≠ f(x) and f(-x) ≠ -f(x), so it's neither even nor odd.
29. What is the x-intercept of f(x) = e^x - 4?
a) ln(4)
b) e⁴
c) 4
d) 0
Answer: a) ln(4)
Solution: Set f(x) = 0 and solve:
e^x - 4 = 0
e^x = 4
x = ln(4)
30. Which function represents a vertical reflection and horizontal stretch by factor 2 of f(x)?
a) -f(x/2)
b) f(-x/2)
c) -f(2x)
d) f(-2x)
Answer: a) -f(x/2)
Solution: -f(x) reflects over x-axis, and f(x/2) stretches horizontally by factor 2.
31. What is the domain of f(x) = arcsin(x)?
a) [-1, 1]
b) (-1, 1)
c) [0, π]
d) (-π/2, π/2)
Answer: a) [-1, 1]
Solution: The arcsine function is defined only for inputs between -1 and 1, inclusive.
32. Which of the following represents an oblique asymptote?
a) lim(x→∞) f(x) = mx + b, where m ≠ 0
b) lim(x→a) f(x) = ∞
c) lim(x→∞) f(x) = b
d) f'(x) = 0
Answer: a) lim(x→∞) f(x) = mx + b, where m ≠ 0
Solution: An oblique asymptote is a slant line that the function approaches as x approaches
infinity.
33. What is the range of f(x) = 3 - e^x?
a) (3, ∞)
b) (-∞, 3)
c) (0, 3)
d) (-∞, ∞)
Answer: b) (-∞, 3)
Solution: As x increases, e^x increases without bound, so 3 - e^x decreases without bound. The
upper limit is 3 (as x approaches -∞).
34. Which transformation represents y = -2f(x - 1) + 3?
a) Reflect over x-axis, stretch vertically by 2, shift right 1, shift up 3
b) Reflect over y-axis, stretch vertically by 2, shift left 1, shift up 3
c) Reflect over x-axis, stretch vertically by 2, shift left 1, shift up 3
d) Reflect over y-axis, compress vertically by 1/2, shift right 1, shift down 3
Answer: c) Reflect over x-axis, stretch vertically by 2, shift left 1, shift up 3
Solution: -2f(x) reflects over x-axis and stretches vertically by 2, (x-1) shifts right by 1, and +3 shifts
up by 3.
8. Which transformation represents y = -f(x)?
a) Reflection over x-axis
b) Reflection over y-axis
c) Vertical stretch by factor of -1
d) Both a and c
Answer: d) Both a and c
Solution: Multiplying f(x) by -1 both reflects the graph over the x-axis and stretches it vertically by a
factor of -1.
9. What is the slope of the line passing through (2, 5) and (4, 9)?
a) 2
b) 4
c) 1
d) 3
Answer: a) 2
Solution: Using the slope formula:
m = (y₂ - y₁) / (x₂ - x₁) = (9 - 5) / (4 - 2) = 4 / 2 = 2
10. Which function represents a horizontal compression by a factor of 3 of f(x)?
a) 3f(x)
b) f(3x)
c) f(x/3)
d) f(x) + 3
Answer: b) f(3x)
Solution: Multiplying x by a constant greater than 1 inside the function compresses the graph
horizontally.
11. What is the vertex of the parabola y = (x - 2)² - 3?
a) (2, -3)
b) (-2, -3)
c) (2, 3)
d) (-2, 3)
Answer: a) (2, -3)
Solution: For y = (x - h)² + k, the vertex is (h, k). Here, h = 2 and k = -3.
12. Which of the following represents an inverse function of f(x) = 2x + 3?
a) f⁻¹(x) = (x - 3)/2
b) f⁻¹(x) = 2x - 3
c) f⁻¹(x) = (x + 3)/2
d) f⁻¹(x) = -2x + 3
Answer: a) f⁻¹(x) = (x - 3)/2
Solution: To find the inverse, swap x and y, then solve for y:
y = 2x + 3
x = 2y + 3
x - 3 = 2y
(x - 3)/2 = y
13. What is the domain of f(x) = 1 / (x² - 4)?
a) All real numbers
b) All real numbers except 2 and -2
c) All real numbers except 0
d) All positive real numbers
Answer: b) All real numbers except 2 and -2
Solution: The denominator cannot be zero:
x² - 4 ≠ 0
(x + 2)(x - 2) ≠ 0
x ≠ 2 and x ≠ -2
14. Which of the following is an odd function?
a) f(x) = x⁴
b) f(x) = x³
c) f(x) = x²
d) f(x) = |x|
Answer: b) f(x) = x³
Solution: For an odd function, f(-x) = -f(x). This is true for x³, as (-x)³ = -x³.
15. What is the range of f(x) = √x?
a) [0, ∞)
b) (-∞, 0]
c) (-∞, ∞)
d) [0, 1]
Answer: a) [0, ∞)
Solution: The square root of a non-negative number is always non-negative, and there's no upper
limit.
16. Which transformation represents y = f(x) + 3?
a) Shift 3 units right
b) Shift 3 units left
c) Shift 3 units up
d) Shift 3 units down
Answer: c) Shift 3 units up
Solution: Adding a constant to the entire function shifts the graph vertically upward by that
amount.
17. What is the period of y = cos(x/2)?
a) π
b) 2π
c) 4π
d) π/2
Answer: c) 4π
Solution: The period of cos(kx) is (2π)/k. Here, k = 1/2, so the period is 2π/(1/2) = 4π.
18. Which function represents a reflection of f(x) over the y-axis?
a) -f(x)
b) f(-x)
c) -f(-x)
d) f(x + 1)
Answer: b) f(-x)
Solution: Negating x inside the function reflects the graph over the y-axis.
19. What is the y-intercept of f(x) = 2x² - 3x + 5?
a) 2
b) -3
c) 5
d) 0
Answer: c) 5
Solution: The y-intercept is f(0):
f(0) = 2(0)² - 3(0) + 5 = 5
20. Which of the following represents a vertical asymptote?
a) lim(x→a) f(x) = ∞
b) lim(x→∞) f(x) = b
c) f(x) = 0
d) f'(x) = 0
Answer: a) lim(x→a) f(x) = ∞
Solution: A vertical asymptote occurs when the function approaches infinity as x approaches a
finite value.
21. What is the slope of the line 3x - 2y = 6?
a) 3/2
b) -3/2
c) 2/3
d) -2/3
Answer: a) 3/2
Solution: Rearrange to slope-intercept form:
-2y = -3x + 6
y = (3/2)x - 3
The coefficient of x is the slope.
22. Which function represents a horizontal reflection and vertical stretch by factor 2 of f(x)?
a) 2f(x)
b) f(2x)
c) 2f(-x)
d) -2f(x)
Answer: c) 2f(-x)
Solution: f(-x) reflects over y-axis, and multiplying by 2 stretches vertically.
23. What is the domain of f(x) = log₂(x - 3)?
a) (3, ∞)
b) [3, ∞)
c) (-∞, 3)
d) (-∞, 3]
Answer: a) (3, ∞)
Solution: The argument of log must be positive:
x - 3 > 0
x > 3
24. Which of the following represents a horizontal asymptote?
a) lim(x→a) f(x) = ∞
b) lim(x→∞) f(x) = b
c) f(x) = 0
d) f'(x) = 0
Answer: b) lim(x→∞) f(x) = b
Solution: A horizontal asymptote occurs when the function approaches a finite value as x
approaches infinity.
25. What is the range of f(x) = 1 / (x² + 1)?
a) (0, 1]
b) [0, 1)
c) (0, ∞)
d) (-∞, ∞)
Answer: a) (0, 1]
Solution: The denominator is always positive and at least 1, so 0 < f(x) ≤ 1, with f(x) = 1 when x = 0.
26. Which transformation represents y = f(2x + 1)?
a) Shift 1 unit left and stretch horizontally by factor 1/2
b) Shift 1 unit right and stretch horizontally by factor 1/2
c) Shift 1/2 unit left and compress horizontally by factor 2
d) Shift 1/2 unit right and compress horizontally by factor 2
Answer: c) Shift 1/2 unit left and compress horizontally by factor 2
Solution: First, factor out the 2: f(2(x + 1/2)). This shows a horizontal compression by 2 and a shift
left by 1/2.
27. What is the period of y = tan(3x)?
a) π/3
b) π/6
c) 2π/3
d) π
Answer: a) π/3
Solution: The period of tan(kx) is π/k. Here, k = 3, so the period is π/3.
28. Which of the following is neither even nor odd?
a) f(x) = x⁴
b) f(x) = x³
c) f(x) = x² + x
d) f(x) = |x|
Answer: c) f(x) = x² + x
Solution: For this function, f(-x) ≠ f(x) and f(-x) ≠ -f(x), so it's neither even nor odd.
29. What is the x-intercept of f(x) = e^x - 4?
a) ln(4)
b) e⁴
c) 4
d) 0
Answer: a) ln(4)
Solution: Set f(x) = 0 and solve:
e^x - 4 = 0
e^x = 4
x = ln(4)
30. Which function represents a vertical reflection and horizontal stretch by factor 2 of f(x)?
a) -f(x/2)
b) f(-x/2)
c) -f(2x)
d) f(-2x)
Answer: a) -f(x/2)
Solution: -f(x) reflects over x-axis, and f(x/2) stretches horizontally by factor 2.
31. What is the domain of f(x) = arcsin(x)?
a) [-1, 1]
b) (-1, 1)
c) [0, π]
d) (-π/2, π/2)
Answer: a) [-1, 1]
Solution: The arcsine function is defined only for inputs between -1 and 1, inclusive.
32. Which of the following represents an oblique asymptote?
a) lim(x→∞) f(x) = mx + b, where m ≠ 0
b) lim(x→a) f(x) = ∞
c) lim(x→∞) f(x) = b
d) f'(x) = 0
Answer: a) lim(x→∞) f(x) = mx + b, where m ≠ 0
Solution: An oblique asymptote is a slant line that the function approaches as x approaches
infinity.
33. What is the range of f(x) = 3 - e^x?
a) (3, ∞)
b) (-∞, 3)
c) (0, 3)
d) (-∞, ∞)
Answer: b) (-∞, 3)
Solution: As x increases, e^x increases without bound, so 3 - e^x decreases without bound. The
upper limit is 3 (as x approaches -∞).
34. Which transformation represents y = -2f(x - 1) + 3?
a) Reflect over x-axis, stretch vertically by 2, shift right 1, shift up 3
b) Reflect over y-axis, stretch vertically by 2, shift left 1, shift up 3
c) Reflect over x-axis, stretch vertically by 2, shift left 1, shift up 3
d) Reflect over y-axis, compress vertically by 1/2, shift right 1, shift down 3
Answer: c) Reflect over x-axis, stretch vertically by 2, shift left 1, shift up 3
Solution: -2f(x) reflects over x-axis and stretches vertically by 2, (x-1) shifts right by 1, and +3 shifts
up by 3.
8. Which transformation represents y = -f(x)?
a) Reflection over x-axis
b) Reflection over y-axis
c) Vertical stretch by factor of -1
d) Both a and c
Answer: d) Both a and c
Solution: Multiplying f(x) by -1 both reflects the graph over the x-axis and stretches it vertically by a
factor of -1.
9. What is the slope of the line passing through (2, 5) and (4, 9)?
a) 2
b) 4
c) 1
d) 3
Answer: a) 2
Solution: Using the slope formula:
m = (y₂ - y₁) / (x₂ - x₁) = (9 - 5) / (4 - 2) = 4 / 2 = 2
10. Which function represents a horizontal compression by a factor of 3 of f(x)?
a) 3f(x)
b) f(3x)
c) f(x/3)
d) f(x) + 3
Answer: b) f(3x)
Solution: Multiplying x by a constant greater than 1 inside the function compresses the graph
horizontally.
11. What is the vertex of the parabola y = (x - 2)² - 3?
a) (2, -3)
b) (-2, -3)
c) (2, 3)
d) (-2, 3)
Answer: a) (2, -3)
Solution: For y = (x - h)² + k, the vertex is (h, k). Here, h = 2 and k = -3.
12. Which of the following represents an inverse function of f(x) = 2x + 3?
a) f⁻¹(x) = (x - 3)/2
b) f⁻¹(x) = 2x - 3
c) f⁻¹(x) = (x + 3)/2
d) f⁻¹(x) = -2x + 3
Answer: a) f⁻¹(x) = (x - 3)/2
Solution: To find the inverse, swap x and y, then solve for y:
y = 2x + 3
x = 2y + 3
x - 3 = 2y
(x - 3)/2 = y
13. What is the domain of f(x) = 1 / (x² - 4)?
a) All real numbers
b) All real numbers except 2 and -2
c) All real numbers except 0
d) All positive real numbers
Answer: b) All real numbers except 2 and -2
Solution: The denominator cannot be zero:
x² - 4 ≠ 0
(x + 2)(x - 2) ≠ 0
x ≠ 2 and x ≠ -2
14. Which of the following is an odd function?
a) f(x) = x⁴
b) f(x) = x³
c) f(x) = x²
d) f(x) = |x|
Answer: b) f(x) = x³
Solution: For an odd function, f(-x) = -f(x). This is true for x³, as (-x)³ = -x³.
15. What is the range of f(x) = √x?
a) [0, ∞)
b) (-∞, 0]
c) (-∞, ∞)
d) [0, 1]
Answer: a) [0, ∞)
Solution: The square root of a non-negative number is always non-negative, and there's no upper
limit.
16. Which transformation represents y = f(x) + 3?
a) Shift 3 units right
b) Shift 3 units left
c) Shift 3 units up
d) Shift 3 units down
Answer: c) Shift 3 units up
Solution: Adding a constant to the entire function shifts the graph vertically upward by that
amount.
17. What is the period of y = cos(x/2)?
a) π
b) 2π
c) 4π
d) π/2
Answer: c) 4π
Solution: The period of cos(kx) is (2π)/k. Here, k = 1/2, so the period is 2π/(1/2) = 4π.
18. Which function represents a reflection of f(x) over the y-axis?
a) -f(x)
b) f(-x)
c) -f(-x)
d) f(x + 1)
Answer: b) f(-x)
Solution: Negating x inside the function reflects the graph over the y-axis.
19. What is the y-intercept of f(x) = 2x² - 3x + 5?
a) 2
b) -3
c) 5
d) 0
Answer: c) 5
Solution: The y-intercept is f(0):
f(0) = 2(0)² - 3(0) + 5 = 5
20. Which of the following represents a vertical asymptote?
a) lim(x→a) f(x) = ∞
b) lim(x→∞) f(x) = b
c) f(x) = 0
d) f'(x) = 0
Answer: a) lim(x→a) f(x) = ∞
Solution: A vertical asymptote occurs when the function approaches infinity as x approaches a
finite value.
21. What is the slope of the line 3x - 2y = 6?
a) 3/2
b) -3/2
c) 2/3
d) -2/3
Answer: a) 3/2
Solution: Rearrange to slope-intercept form:
-2y = -3x + 6
y = (3/2)x - 3
The coefficient of x is the slope.
22. Which function represents a horizontal reflection and vertical stretch by factor 2 of f(x)?
a) 2f(x)
b) f(2x)
c) 2f(-x)
d) -2f(x)
Answer: c) 2f(-x)
Solution: f(-x) reflects over y-axis, and multiplying by 2 stretches vertically.
23. What is the domain of f(x) = log₂(x - 3)?
a) (3, ∞)
b) [3, ∞)
c) (-∞, 3)
d) (-∞, 3]
Answer: a) (3, ∞)
Solution: The argument of log must be positive:
x - 3 > 0
x > 3
24. Which of the following represents a horizontal asymptote?
a) lim(x→a) f(x) = ∞
b) lim(x→∞) f(x) = b
c) f(x) = 0
d) f'(x) = 0
Answer: b) lim(x→∞) f(x) = b
Solution: A horizontal asymptote occurs when the function approaches a finite value as x
approaches infinity.
25. What is the range of f(x) = 1 / (x² + 1)?
a) (0, 1]
b) [0, 1)
c) (0, ∞)
d) (-∞, ∞)
Answer: a) (0, 1]
Solution: The denominator is always positive and at least 1, so 0 < f(x) ≤ 1, with f(x) = 1 when x = 0.
26. Which transformation represents y = f(2x + 1)?
a) Shift 1 unit left and stretch horizontally by factor 1/2
b) Shift 1 unit right and stretch horizontally by factor 1/2
c) Shift 1/2 unit left and compress horizontally by factor 2
d) Shift 1/2 unit right and compress horizontally by factor 2
Answer: c) Shift 1/2 unit left and compress horizontally by factor 2
Solution: First, factor out the 2: f(2(x + 1/2)). This shows a horizontal compression by 2 and a shift
left by 1/2.
27. What is the period of y = tan(3x)?
a) π/3
b) π/6
c) 2π/3
d) π
Answer: a) π/3
Solution: The period of tan(kx) is π/k. Here, k = 3, so the period is π/3.
28. Which of the following is neither even nor odd?
a) f(x) = x⁴
b) f(x) = x³
c) f(x) = x² + x
d) f(x) = |x|
Answer: c) f(x) = x² + x
Solution: For this function, f(-x) ≠ f(x) and f(-x) ≠ -f(x), so it's neither even nor odd.
29. What is the x-intercept of f(x) = e^x - 4?
a) ln(4)
b) e⁴
c) 4
d) 0
Answer: a) ln(4)
Solution: Set f(x) = 0 and solve:
e^x - 4 = 0
e^x = 4
x = ln(4)
30. Which function represents a vertical reflection and horizontal stretch by factor 2 of f(x)?
a) -f(x/2)
b) f(-x/2)
c) -f(2x)
d) f(-2x)
Answer: a) -f(x/2)
Solution: -f(x) reflects over x-axis, and f(x/2) stretches horizontally by factor 2.
31. What is the domain of f(x) = arcsin(x)?
a) [-1, 1]
b) (-1, 1)
c) [0, π]
d) (-π/2, π/2)
Answer: a) [-1, 1]
Solution: The arcsine function is defined only for inputs between -1 and 1, inclusive.
32. Which of the following represents an oblique asymptote?
a) lim(x→∞) f(x) = mx + b, where m ≠ 0
b) lim(x→a) f(x) = ∞
c) lim(x→∞) f(x) = b
d) f'(x) = 0
Answer: a) lim(x→∞) f(x) = mx + b, where m ≠ 0
Solution: An oblique asymptote is a slant line that the function approaches as x approaches
infinity.
33. What is the range of f(x) = 3 - e^x?
a) (3, ∞)
b) (-∞, 3)
c) (0, 3)
d) (-∞, ∞)
Answer: b) (-∞, 3)
Solution: As x increases, e^x increases without bound, so 3 - e^x decreases without bound. The
upper limit is 3 (as x approaches -∞).
34. Which transformation represents y = -2f(x - 1) + 3?
a) Reflect over x-axis, stretch vertically by 2, shift right 1, shift up 3
b) Reflect over y-axis, stretch vertically by 2, shift left 1, shift up 3
c) Reflect over x-axis, stretch vertically by 2, shift left 1, shift up 3
d) Reflect over y-axis, compress vertically by 1/2, shift right 1, shift down 3
Answer: c) Reflect over x-axis, stretch vertically by 2, shift left 1, shift up 3
Solution: -2f(x) reflects over x-axis and stretches vertically by 2, (x-1) shifts right by 1, and +3 shifts
up by 3.
35. What is the period of y = sec(x/4)?
a) 4π
b) 8π
c) π/2
d) π/4
Answer: b) 8π
