Graphs of Functions
Lesson 3
Warm Up – Perform the Operations and
Simplify
4
a. 12 
x2
3
x
b. 2
 2
x  x  20 x  4 x  5
2 x 3  11x 2  6 x
x  10
c.
 2
5x
2 x  5x  3
x7
x7
d.

2x  9 2x  9
4
a. 12 
x2

Solution
4
12 
x2
12x  2
4


x2
x2
12 x  24  4

x2
12 x  20

x2
3
x
b. 2
 2
x  x  20 x  4 x  5

x

Solution 
x  5x  4 x  5x  1
3 x  1  x x  4 

x  5x  4x  1
3
3x  3  x 2  4 x

x  5x  4x  1
x  x3

x  5x  4x  1
2
2 x 3  11x 2  6 x
x  10
c.
 2
5x
2 x  5x  3

Solution
2 x 3  11x 2  6 x
x  10
 2
5x
2 x  5x  3
x 2 x 2  11x  6
x  10


2 x  1x  3
5x
x2 x  1 x  6  x  10 

5 x2 x  1 x  3

x  6  x  10 

5 x  3


x7
x7
d.

2x  9 2x  9

Solution
x7
x7

2x  9 2x  9
x  7 2x  9


2x  9 x  7
x7

x7
Domain & Range of a Function
What is the
domain of
the graph of
the function
f?
A :  1,4
Domain & Range of a Function
What is the
range of
the graph of
the function
f?
 5,4
Domain & Range of a Function
Find f  1 and f 2.
f  1  5
f 2  4

Let’s look at domain and range of a
function using an algebraic approach.

Then, let’s check it with a graphical
approach.
Find the domain and range of
f x   x  4.

Algebraic Approach
The expression under the radical can not be negative.
Therefore, x  4  0.
Domain
A: x  4
Since the domain is never negative the
or
range is the set of all nonnegative real
4,   numbers.
A: y  0
or
0,  
Range
Find the domain and range of
f x   x  4.

Graphical Approach
Increasing and Decreasing
Functions

The more you know about the graph of
a function, the more you know about
the function itself.

Consider the graph on the next slide.
Falls from x = -2 to x = 0.
Is constant from x = 0 to
x = 2.
Rises from x = 2 to x = 4.
Ex: Find the open intervals on which the
function is increasing, decreasing, or constant.
Increases over
the entire real
line.
Ex: Find the open intervals on which the
function is increasing, decreasing, or constant.
INCREASING :
 ,1 and 1,  
DECREASING :
 1,1
Ex: Find the open intervals on which the
function is increasing, decreasing, or constant.
INCREASING :
 ,0
CONSTANT :
0,2
DECREASING :
2. 
Relative Minimum and
Maximum Values

The point at which a function changes
its increasing, decreasing, or constant
behavior are helpful in determining the
relative maximum or relative
minimum values of a function.
General Points – We’ll find
EXACT points later……
Approximating a Relative
Minimum

Example: Use a GDC to approximate
the relative minimum of the function
given by
f x   3x  4 x  2.
2
f x   3x  4 x  2.
2


Put the function into the “y = “ the
press zoom 6 to look at the graph.
Press trace to follow the line to the
lowest point.
Example

Use a GDC to approximate the relative
minimum and relative maximum of the
function given by
f x    x  x.
3
Solution
Relative Minimum
(-0.58, -0.38)
Solution
Relative Maximum
(0.58, 0.38)
Step Functions and
Piecewise-Defined Functions
Because of the vertical jumps, the greatest integer function is an example
of a step
function.
Let’s graph a PiecewiseDefined Function

Sketch the graph of
 2 x  3, x  1
f x   
 x  4, x  1
Notice when open
dots and closed
dots are used. Why?
Even and Odd Functions
Graphically
Algebraically
Let’s look at the graphs again and see if this applies.
Graphically
☺
☺
Example

Determine whether each function is
even, odd, or neither.
Algebraic
Graphical –
Symmetric to
Origin
Algebraic
Graphical –
Symmetric to yaxis
Algebraic
Graphical – NOT
Symmetric to
origin OR y-axis.
You Try

Is the function
f x   x

Even, Odd, of Neither?
Solution
f x   x
Symmetric about the y-axis.