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