1.2 Functions

• Determine whether relations between two variables represent functions

• Use function notation and evaluate functions

• Find the domains of functions• Use functions to model and solve real-life

problems• Evaluate difference quotients

Definition of a Function:A function is a relation in which each element of the domain

(the set of x-values, or input) is mapped to one and only one element of the range (the set of y-values, or output).

Function Not a Function One-to-one

Function

A Function can be represented several ways:

• Verbally – by a sentence that states how the input is related to the output.

• Numerically – in the form of a table or a list of ordered pairs.

• Graphically – a set of points graphed on the x-y coordinate plane.

• Algebraically – by an equation in two variables.

Example 1

Input x 2 2 3 4 5

Output y 11 10 8 5 1

Example 2

Which of the equations represents y as a function of x?

a. b. x y2 1 x y 2 1

Example 3Let g x x x( ) 2 4 1

g(2)=

g(t)=

g(x+2)=

Example 4 : Evaluate the piecewise function when x=-1 and x=0.

{ ,

{ ,

x x

x x

2 1 0

1 0

Example 5 : Find the domain of each function

a. f: {(-3,0),(-1,4),(0,2),(2,2),(4,-1)}

b.

c.

d.

e.

3 4 52x x

h xx

( ) 1

5

V r 4

33

k x x( ) 4 3

Example 6

Use a graphing calculator to find the domain and range of the function

f x x( ) 9 2

Example 7The number N (in millions) of cellular phone subscribers in the United

States increased in a linear pattern from 1995 to 1997, as shown on p.22. Then, in 1998, the number of subscribers took a jump, and until 2001, increased in a different linear pattern. These two patterns can be approximated by the function

where t represents the year, with

t=5 corresponding to 1995. Use this function to approximate the number of cellular phone subscribers for each year from 1995 to 2001.

N tt t

t t( ) {

. . ,

. . ,

1 0 7 5 2 0 1 5 7

2 0 11 9 2 8 8 11

Example 8

A baseball is hit at a point 3 feet above the ground at a velocity of 100 feet per second and an angle of 45 degrees. The path of the baseball is given by the function where y and x are measured in feet. Will the baseball clear a 10 foot fence located 300 feet from home plate?

f x x x( ) . 0 0 3 2 32

Student Example

A baseball is hit at a point 4 feet above the ground at a velocity of 120 feet per second and an angle of 45 degrees. The path of the baseball is given by the function where y and x are measured in feet. Will the baseball clear an 8 foot fence located 350 feet from home plate?

f x x x( ) . 0 0 3 8 42

Example 9

For .f x x x findf x h f x

h( ) ,

( ) ( )

2 4 7

Student ExampleEvaluate for

f(-3)

f(x+1)

f(x+h)-f(x)

f x x x( ) 2 3 2