Algebra Test

College Algebra MATH 107 Fall, 2019, V1.5

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MATH 107 FINAL EXAMINATION

This is an open-book exam. You may refer to your text and other course materials as you work

on the exam, and you may use a calculator. You must complete the exam individually.

Neither collaboration nor consultation with others is allowed.

Record your answers and work on the separate answer sheet provided.

There are 30 problems.

Problems #1–12 are Multiple Choice.

Problems #13–21 are Short Answer. (Work not required to be shown)

Problems #22–30 are Short Answer with work required to be shown.

MULTIPLE CHOICE

1. Determine the domain and range of the piecewise function. 1. ______

A. Domain [–1, 1]; Range [–1, 1]

B. Domain [–1, 3]; Range [–3, 1]

C. Domain [–3, 1]; Range [–1, 3]

D. Domain [1/2, 3/2]; Range [0, 1]

2. Solve: 7 4 3x x− − = + 2. ______

A. −2

B. −8, −2

C. −1/2

D. No solution

2 4 -4

-2

-4

2

4

-2

College Algebra MATH 107 Fall, 2019, V1.5

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3. Determine the interval(s) on which the function is decreasing. 3. ______

A. (–, – 2.5) and (1, 4.5)

B. (– 1, 3)

C. (– 2, 2)

D. (–, 2)

4. Determine whether the graph of 3 xy = is symmetric with respect to the origin,

the x-axis, or the y-axis. 4. ______

A. symmetric with respect to the x-axis only

B. symmetric with respect to the y-axis only

C. symmetric with respect to the origin only

D. not symmetric with respect to the x-axis, not symmetric with respect to the y-axis, and

not symmetric with respect to the origin

5. Solve, and express the answer in interval notation: | 5 – 6x |  13. 5. ______

A. (–, −4/3]  [3, )

B. [4/3, )

C. [–4/3, 3]

D. (–, 3]  [−4/3, )

College Algebra MATH 107 Fall, 2019, V1.5

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6. Which of the following represents the graph of 9x − 4y = 36 ? 6. ______

A. B.

C. D.

College Algebra MATH 107 Fall, 2019, V1.5

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7. Write a slope-intercept equation for a line parallel to the line x – 4y = 6 which passes through

the point (12, –3). 7. ______

A. 4 45y x= − +

B. 1

1 4

y x= − −

C. 1

3 4

y x= −

D. 1

6 4

y x= −

8. Does the graph below represent a function and is it one-to-one? 8. ______

A. It is a function but not one-to-one.

B. It is not a function but it is one-to-one.

C. It is a function and it is one-to-one.

D. It is not a function and it is not one-to-one.

College Algebra MATH 107 Fall, 2019, V1.5

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9. Write as an equivalent expression: 3 log (x + 4) – 2 log y + log 1 9. ______

A. 3

2

64 log

x

y

 +    

B. ( )

3

2

4 log

x

y

 +      

C. 12 log( )

log(2 )

x

y

D. ( )log 3 13 2x y+ −

10. Which of the functions corresponds to the graph? 10. ______

A. ( ) 1xf x e−= −

B. ( ) 1 xf x e= −

C. ( ) 2 xf x e= −

D. ( ) xf x e= −

College Algebra MATH 107 Fall, 2019, V1.5

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11. Suppose that for a function f , the equation f (x) = 0 has exactly one real-number solution.

Which of the following statements MUST be true? 11. ______

A. f has exactly one x-intercept.

B. f is a one-to-one function.

C. f is a linear function.

D. f is an invertible function.

12. The graph of y = f (x) is shown at the left and the graph of y = g(x) is shown at the right. (No

formulas are given.) What is the relationship between g(x) and f (x)?

12. ______

y = f (x) y = g(x)

A. g(x) = f (x – 2)

B. g(x) = f (x + 2)

C. g(x) = f (x – 4)

D. g(x) = f (x) – 2

-2 2 4 -4

-2

-4

2

4

-2 2 4 -4

-2

-4

2

4

College Algebra MATH 107 Fall, 2019, V1.5

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SHORT ANSWER:

13. Multiply and simplify: (6 + 5i)(3 − 8i).

Write the answer in the form a + bi, where a and b are real numbers. Answer: ________

14. Solve, and write the answer in interval notation: 3

0 1

x

x

+ 

+ . Answer: ________

15. Water initially at 200 F. is left in a room of temperature 70 F to cool.

After t minutes, the temperature T of the water is given by T(t) = 70 + 130e – 0.096 t

Find the temperature of the water 20 minutes after it is left to cool. (Round to the nearest degree.)

Answer: ________

16. Find the value of the logarithm: 5 1

log 125

     

. Answer: ________

17. Solve: 3 72 4x+ = . Answer: ________

18. Suppose $3,200 is invested in an account at an annual interest rate of 5.7% compounded

continuously. How long (to the nearest tenth of a year) will it take the investment to double in

size? Answer: ________

19. Let f (x) = x2 + 20x + 96.

(a) Find the vertex. Answer: ________

(b) State the range of the function. Answer: ________

(c) On what interval is the function decreasing? Answer: ________

College Algebra MATH 107 Fall, 2019, V1.5

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20. Consider the polynomial P(x), shown in both standard form and factored form.

3 21 5 1 ( ) 3 ( 1)( 3)( 2)

2 2 2 P x x x x x x x= + − − = + + −

(a) Which sketch illustrates the end behavior of the polynomial function?

Answer: ________

(b) State the y-intercept. Answer: ________________

(c) State the zeros of the function. Answer: ________________

(d) State which graph below is the graph of P(x). Answer: ________

GRAPH A

GRAPH B

GRAPH C

GRAPH D

A.

B. C. D.

vvvv

vvvv

vvvv vvvv

College Algebra MATH 107 Fall, 2019, V1.5

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21. Let 3 9

( ) 2

x f x

x

− =

− .

(a) State the domain. Answer: _________________

(b) State the horizontal asymptote. Answer: _________________

(c) State the vertical asymptote(s). Answer: _________________

(d) Which of the following represents the graph of 3 9

( ) 2

x f x

x

− =

− ? Answer: ______________

GRAPH A. GRAPH B.

GRAPH C. GRAPH D.

College Algebra MATH 107 Fall, 2019, V1.5

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SHORT ANSWER, with work required to be shown, as indicated.

22. Let 2)( += xxf and g(x) = x – 5 .

(a) Find )1(− 

  



g

f . Show work.

(b) Find the domain of the quotient function g

f . Explain.

23. Points (3, 2) and (–5, 6) are endpoints of the diameter of a circle.

(a) What is the length of the diameter? Give the exact answer, simplified as much as possible.

Show work.

(b) What is the center point C of the circle?

(c) Given the point C you found in part (b), state the point symmetric to C about the x-axis.

24. Find the equation for a line which passes through the points (2, 1) and (6, –7). Write the

equation in slope-intercept form. Show work.

25. A salesperson earns a base salary of $1,560 per month and a commission of 7.8% on the

amount of sales. If the salesperson has a paycheck of $4,543.50 for one month, what was the

amount of sales for the month? Show work.

26. Let f (x) = 5x2 – 4 and g(x) = x – 3.

(a) Find the composite function ))(( xgf  and simplify. Show work.

(b) Find ( ) (2)f go . Show work.

27. Find the exact solutions and simplify as much as possible: 3x2 + 1 = 6x. Show work.

28. Given the function 1

( ) 8 6

f x x= + , find a formula for the inverse function. Show work.

College Algebra MATH 107 Fall, 2019, V1.5

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29. Donut Delights, Inc. has determined that when x donuts are made daily, the profit P, in

dollars, is given by

P(x) = –0.001 x2 + 1.9x – 350

(a) What is the company’s profit if 400 donuts are made daily?

(b) How many donuts should be made daily in order to maximize the company’s profit? Show

work.

30. Solve: 2

9 36 0

6 36

x

x x

+ + =

+ − . Show work.