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. x 2  y  1
b.  x  y 2  1
Example 3
Let g( x)   x
g(2)=
g(t)=
g(x+2)=
2
 4x  1
Example 4 : Evaluate the piecewise function when x=-1
and x=0.
{x  1, x 0
{x  1, x  0
2
Example 5 : Find the domain of each function
a.
f: {(-3,0),(-1,4),(0,2),(2,2),(4,-1)}
b.
 3x 2  4 x  5
c.
1
h( x ) 
x 5
d.
4
V   r3
3
e.
k ( x)  4  3x
Example 6
Use a graphing calculator to find the domain and range of the function
f ( x)  9  x 2
Example 7
The 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
10.75t  201
. ,5  t  7
approximated by the function
N (t )  {
where t represents the year, with
2011
. t  92.8,8  t  11
t=5 corresponding to 1995. Use this function to approximate the
number of cellular phone subscribers for each year from 1995 to 2001.
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 f ( x)   .0032 x 2  x  3 where y and x are
measured in feet. Will the baseball clear a 10 foot fence located 300
feet from home plate?
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 f ( x)   .0038x 2  x  4 where y and x are
measured in feet. Will the baseball clear an 8 foot fence located 350
feet from home plate?
Example 9
For f ( x)  x 2  4 x  7, find
f ( x  h)  f ( x )
.
h
Student Example
Evaluate f ( x)  2  3x  x 2 for
f(-3)
f(x+1)
f(x+h)-f(x)