college algebra

Module 1:  Functions

Functions  

Consider the equation: y = 2x + 3, some ordered pairs that satisfy this equation are: (-2,-1), (-1,1), (0,3).

Dependent and Independent Variables :

In this equation: y = 2x +3, the value of  y depends on the value of  x, then “y” is the dependent variable and “x” is the independent variable.

Relation, Domain, and Range:

A  relation is any set of ordered pairs of the form (x, y). The of x-coordinates is called the  domain of the relation. The set of y-coordinates is called the  range of the relation

Function:

Is a relation in which each element of the domain corresponds to exactly one element in the range.

Example: Give the domain and range, then determine whether the relation is a function.

A. {(1,4), (2,3), (3,5), (-1,3), (0,6)}

B. {(-1,3), (4,2), (3,1), (2,6), (3,5)}

Solution:

A. The domain is {1,2,3,-1,0} and the range is {4,3,5,6}. The number 3 was included only once, even though appears in both pairs (2,3) and (-1,3). Since no x-value corresponds to more than one y-value in the range, this relation is a  function.

B. The domain is {-1,4,3,2} and the range is {3,2,1,6,5}. Since the ordered pairs (3,1) and (3,5) have the same first coordinate (x) and a different second coordinate, each value does not correspond to exactly one value in the range; therefore, this relation is  not a function.

From this example, an alternate definition of  function is a set of ordered pairs in which no first coordinate is repeated.

Composite Function

Given the function: f[g(x)] = (f o  g)(x)

Function  f[g(x)]   is called a  composition of  f with  g  or the  composite function of  f with  g

Example:

Given f(x) = x – 3 and g(x) = x + 7, find:

a) (f o  g)(x)

b) (f o  g)(2)

c) (g o  f)(x)

d) (g o  f)(2)

Solution:

a) (f o  g)(x) 

since f(x) = x + 4, then (f o  g)(x) = f[g(x)] = (x + 7) – 3 = x + 4

(f o  g)(x) = x + 4

b) (f o  g)(2) = x + 4 = 2 + 4 = 6

(f o  g)(2) = 6

c) (g o  f)(x)

since g(x) = x + 7, then (g o  f)(x) = g[f(x)] = (x – 3) + 7 = x + 4

(g o  f)(x) = x + 4

d) (g o  f)(2) = x + 4 = 2 +4 = 6

(g o  f)(2) = 6

Inverse Function

One-to-one functions consist of ordered pairs in which each x-coordinate corresponds to exactly one y-coordinate and each y-coordinate corresponds to exactly one x-coordinate. Such a relationship allows creating  new functions called the  inverse function. Only one-to-one functions have inverses.

Examples:

Function f(x): {(0,1), (1,4), (-3,-2)}

The inverse function will be: f−1(x)= {(1,0), (4,1), (-2,-3)

Function f(x) = 4x + 2, this function also can be expressed as: 

y = 4x + 2

Interchange x and y:

x = 4y +2, solving for y:

x – 2 = 4y

x−24=y

The inverse will be: f−1(x) = y

x−24=y

Linear Function or Linear Equation

A linear function is a function of the form f(x) = mx + b or y = mx + b

· The graph of any linear equation is a straight line.

· The domain of any linear function is all real numbers.

· If m ≠ 0, then the range of any linear function is all real numbers.

· m, is the slope of the line (is a measure of the steepness of a line)

· b, is the y-intercept

Linear Equations (mathsisfun.com)Links to an external site.

Example: 

Find the linear equation that passes through the points (2,3) and (1,4) steps: 1) find the slope (m), m = y2−y1x2−x1

let (2, 3) be (x1,y1) and (1, 4) be (x2,y2)

 m=y2−y1x2−x1=4−31−2=1−1=−1

2) find the point-slope form of the equation of the line

y–y1=m(x–x1), replacing the values of m, x1 , and y1 [ordered pair (2,3)]

y – 3 = - 1 (x – 2)

y – 3 = -x + 2

y = -x + 2 + 3

y = - x + 5

References

Angel, A., & Rundle, D. (2018).  Intermediate algebra for college students. (10th edition). Pearson.

Module 1:  Functions

Functions  

Consider the equation: y = 2x + 3, some ordered pairs that satisfy this equation are: (-2,-1), (-1,1), (0,3).

Dependent and Independent Variables :

In this equation: y = 2x +3, the value of  y depends on the value of  x, then “y” is the dependent variable and “x” is the independent variable.

Relation, Domain, and Range:

A  relation is any set of ordered pairs of the form (x, y). The of x-coordinates is called the  domain of the relation. The set of y-coordinates is called the  range of the relation

Function:

Is a relation in which each element of the domain corresponds to exactly one element in the range.

Example: Give the domain and range, then determine whether the relation is a function.

A. {(1,4), (2,3), (3,5), (-1,3), (0,6)}

B. {(-1,3), (4,2), (3,1), (2,6), (3,5)}

Solution:

A. The domain is {1,2,3,-1,0} and the range is {4,3,5,6}. The number 3 was included only once, even though appears in both pairs (2,3) and (-1,3). Since no x-value corresponds to more than one y-value in the range, this relation is a  function.

B. The domain is {-1,4,3,2} and the range is {3,2,1,6,5}. Since the ordered pairs (3,1) and (3,5) have the same first coordinate (x) and a different second coordinate, each value does not correspond to exactly one value in the range; therefore, this relation is  not a function.

From this example, an alternate definition of  function is a set of ordered pairs in which no first coordinate is repeated.

Composite Function

Given the function: f[g(x)] = (f o  g)(x)

Function  f[g(x)]   is called a  composition of  f with  g  or the  composite function of  f with  g

Example:

Given f(x) = x – 3 and g(x) = x + 7, find:

a) (f o  g)(x)

b) (f o  g)(2)

c) (g o  f)(x)

d) (g o  f)(2)

Solution:

a) (f o  g)(x) 

since f(x) = x + 4, then (f o  g)(x) = f[g(x)] = (x + 7) – 3 = x + 4

(f o  g)(x) = x + 4

b) (f o  g)(2) = x + 4 = 2 + 4 = 6

(f o  g)(2) = 6

c) (g o  f)(x)

since g(x) = x + 7, then (g o  f)(x) = g[f(x)] = (x – 3) + 7 = x + 4

(g o  f)(x) = x + 4

d) (g o  f)(2) = x + 4 = 2 +4 = 6

(g o  f)(2) = 6

Inverse Function

One-to-one functions consist of ordered pairs in which each x-coordinate corresponds to exactly one y-coordinate and each y-coordinate corresponds to exactly one x-coordinate. Such a relationship allows creating  new functions called the  inverse function. Only one-to-one functions have inverses.

Examples:

Function f(x): {(0,1), (1,4), (-3,-2)}

The inverse function will be: f−1(x)= {(1,0), (4,1), (-2,-3)

Function f(x) = 4x + 2, this function also can be expressed as: 

y = 4x + 2

Interchange x and y:

x = 4y +2, solving for y:

x – 2 = 4y

x−24=y

The inverse will be: f−1(x) = y

x−24=y

Linear Function or Linear Equation

A linear function is a function of the form f(x) = mx + b or y = mx + b

· The graph of any linear equation is a straight line.

· The domain of any linear function is all real numbers.

· If m ≠ 0, then the range of any linear function is all real numbers.

· m, is the slope of the line (is a measure of the steepness of a line)

· b, is the y-intercept

Linear Equations (mathsisfun.com)Links to an external site.

Example: 

Find the linear equation that passes through the points (2,3) and (1,4) steps: 1) find the slope (m), m = y2−y1x2−x1

let (2, 3) be (x1,y1) and (1, 4) be (x2,y2)

 m=y2−y1x2−x1=4−31−2=1−1=−1

2) find the point-slope form of the equation of the line

y–y1=m(x–x1), replacing the values of m, x1 , and y1 [ordered pair (2,3)]

y – 3 = - 1 (x – 2)

y – 3 = -x + 2

y = -x + 2 + 3

y = - x + 5

References

Angel, A., & Rundle, D. (2018).  Intermediate algebra for college students. (10th edition). Pearson.