Section P.3 – Functions and
their Graphs
Functions
A relation such that there is no more than one
output for each input
A
W
B
Z
C
Algebraic
Function
Can be written as finite sums, differences,
multiples, quotients, and radicals involving xn.
f x3
x x
1
0
2
Examples:
Transcendental
Function
1
gx2x
4
4x
A function that is not Algebraic.
hx sinx
Examples: gx lnx
4 Examples of Functions
X
Y
10 2
15 -5
X
Y
-3
1
-1
0
0
4
5
7
7
3
18 -5
20 1
7
These are all
functions
because every x
value has only
one possible y
value
-5
Every one of these
functions is a
relation.
3 Examples of Non-Functions
X
Y
0
4
1 10
Not a function
since x=-4
can be either
y=7 or y=1
2 11
1
-3
5
3
Not a function
since x=1 can
be either
y=10 or y=-3
Not a function since
multiple x values
have multiple y
values
Every one of these
non-functions is a
relation.
The Vertical Line Test
If a vertical line
intersects a curve
more than once, it
is not a function.
Use the vertical
line test to decide
which graphs are
functions. Make
sure to circle the
functions.
The Vertical Line Test
If a vertical line
intersects a curve
more than once, it
is not a function.
Use the vertical
line test to decide
which graphs are
functions. Make
sure to circle the
functions.
Function Notation: f(x)
Equations that are functions are typically written in a
different form than “y =.” Below is an example of function
notation:
It does stand for
Does not stand
f  x  x
“plug a value for x
for “f times x”
into a formula f”
The equation above is read:
f of x equals the square root of x.
The first letter, in this case f, is the name of the function
machine and the value inside the parentheses is the
input. The expression to the right of the equal sign
shows what the machine does to the input.
Example
If g(x) = 2x + 3, find g(5).
g 5  2 5  3
When evaluating,
do not write g(x)!
You wanted to
find g(5). So the
complete final
answer includes
g(5) not g(x)
g 5   10  3
g  5   13
You want x=5
since g(x) was
changed to g(5)
Solving v Evaluating
If f
x 

2
3
x  3, com plete the follow ing:
No equal sign
Equal sign
a. E valuate f  3 
x 
b . F in d x if f
Substitute and Evaluate
Solve for x
The input (or x) is 3.
The output is -5.

2
3
3  3
2  3
1
5  
2
3
x3
8  
2
3
x
12  x
5
Number Sets
Natural Numbers: Counting numbers
(maybe 0, 1, 2, 3, 4, and so on)
Integers: Positive and negative counting
numbers (-2, -1, 0, 1, 2, and so on)
Rational Numbers: a number that can be
expressed as an integer fraction
(-3/2, -1/3, 0, 1, 55/7, 22, and so on)
Irrational Numbers: a number that can
NOT be expressed as an integer
fraction (π, √2, and so on)
NONE
Number Sets
Real Numbers: The set of all rational
and irrational numbers
Real Number Venn
Diagram:
Rational Numbers
Integers
Irrational
Natural Numbers
Numbers
Set Notation
Not Included
The interval does NOT include the endpoint(s)
Interval Notation Inequality Notation Graph
Parentheses
( )
< Less than
> Greater than
Open Dot
Included
The interval does include the endpoint(s)
Interval Notation Inequality Notation Graph
Square Bracket
[ ]
≤ Less than
≥ Greater than
Closed Dot
Example 1
Graphically and algebraically represent the following:
All real numbers greater than 11
Graph:
10
Inequality:
Interval:
x1
1
11,
11
12
Infinity never ends.
Thus we always
use parentheses to
indicate there is no
endpoint.
Example 2
Describe, graphically, and algebraically represent
the following:
1

x5
Description: All real numbers greater than or
equal to 1 and less than 5
Graph:
Interval:
1
1,5
3
5
Example 3
Describe and algebraically represent the
following:
-2
1
4
All real numbers less than -2 or
Describe:
greater than 4
Inequality:
Symbolic:
x


2
o
r
x

4
The union or
combination of the
two sets.


,2

4
,




Domain and Range
Domain
All possible input values (usually x),
which allows the function to work.
Range
All possible output values (usually
y), which result from using the
function.
f
x
y
The domain and range help determine the window of a graph.
Example 1
Describe the domain and range of both functions in
interval notation:
y

x1
x

9




, 
Domain: 

8
,2
2
,9


Domain: 
2
5
,


Range: 
Range: 7,8
Example 2
Find the domain and range of h
t
4

3
t.


The domain is not
obvious with the
graph or table.
The input to a square root
function must be greater
than or equal to 0
4

3
t

0

3
t


4
4
Dividing by a
t3
negative switches

the sign
t
-32 -20 -15
5
-4
0
h
10
-7
4
2
8
7
DOMAIN:

1
1
2
3
ER ER

RANGE:
, 

4
3
The range is
clear from
the graph
0, and
table.
Piecewise Functions
For Piecewise Functions, different formulas are
used in different regions of the domain.
Ex: An absolute value function can be written as a
piecewise function:

x ifx0

x
x ifx0
Example 1
Write a piecewise function for each given graph.
gx

f x
x
5 if
ifx
0
1
x


4
gx   1
f x  
0

x
1ifx
7 if
x


4
2



Example 2
Rewrite fx
as a piecewise function.

x

2

1


Use a graph or table to help.
x

-3
f(x) 6
-2
-1
0
1
2
3
4
5
4
3
2
1
2
3
Find the x value of
the vertex
Change the absolute
values to parentheses.
Plus make the one on
the left negative.
2
1ifx

x


2
f x  
x
2
1 ifx

2

 
Basic Types of Transformations
Parent/Original Function: yf x
When negative,
the original graph
is flipped about
the x-axis
A vertical stretch if
|a|>1and a vertical
compression if
|a|<1
Horizontal shift of h
units
y

a

f

x

h

k


When negative, the
original graph is flipped
about the y-axis
Vertical shift of k units
( h, k ): The Key Point
Transformation Example
Use the graph of y  x below to describe and
1
sketch the graph of y
.

3
x

4
1
Description:
Shift the parent
graph four units to
the left and three
units down.
Composition of Functions
Substituting a function or it’s value into another
function. There are two notations:
f gx
Second
OR
First
f gx
(inside parentheses always first)
g
f
Example 1
2
g
x


x
5
Let f
and
. Find:
x
2
x

3

f g1
g
1


1
5


2
Substitute
x=1 into
g(x) first
f
4

2
4

3


1
5
8
3
4
4
11
f
g
1
1
1


Substitute
the result
into f(x)
last
Example 2
2
g
x


x
5
Let f
and
. Find:
x
2
x

3

g f x
x

Substitute the result into g(x) last
g
2
x

3


2
x

3

5




2
Substitute x
into f(x) first
f
x
2
x

3
2
x

3


2
x

3
2
x

3

5






4
x

1
2
x
9

5


2


4
x

1
2
x

9

5
2


4
x1
2
x
4
2
g
f
x


4
x

1
2
x

4




2
Even v Odd Function
Even
Function
Tests…
Symmetrical with
respect to the y-axis.
f x  f
x
Replacing x in the
function by –x yields
an equivalent function.
Odd
Function
Symmetrical with
respect to the origin.
f x   f
x
Replacing x in the
function by –x yields the
opposite of the function.
Example
Is the function f
 x   x odd, even, or neither?
Check out the graph first.
4
Test by Replacing x in the
function by –x.
f x  x
f x  x
4
4
An equivalent
equation.
The equation is even.
Delta x
x stands for “the change in x.” It is a variable that
represents ONE unknown value. For example, if x1 = 5
and x2 = 7 then Δx = 7 – 5 = 2. Δx can be algebraically
manipulated similarly to single letter variables. Simplify
the following statements:
Δ
1
. 5x

1

2
5
x

1
5
x

1




2
5
x

5
x
5
x
1

2
2
5
1
0
x

1
x

2
.
x xx
x
x xx
x
x
x
2
1