2.4 graphs of functions the graph of a function is the graph of its ordered pairs
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2.4 Graphs of Functions
The graph of a function is the graph of its ordered pairs.
Graphing a Function by Plotting Points
feature in your
graphing utility. You can use it to create
tables of coordinates for a function. You
will need to enter the equation of the
function and sepcify the starting value for
x
Use the TABLE
and the increment betweenTblStart
successive -valuesx Tbl .
Steps for Graphing an Equation Using a Graphing Utility
: Solve the equation for in
terms
St
o
ep
.
1
f
y
x
Steps for Graphing an Equation Using a Graphing Utility
: Get in to the graphing mode of your
graphing utility. The screen will usually
display , prompting you to
enter the expression involving that you
St
found i
ep 2
y
x
n step 1.
Steps for Graphing an Equation Using a Graphing Utility
: Select the viewing window.
Adjust the viewing window until
a complete graph is obta
Step 3
ined.
Ex 1: Graphing a Function by Plotting Points
Graph the function: ( ) 2f x x
Hint: To do so, use integer values of x from
the set {0,1,4,9,16,25} to obtain six ordered
pairs. Plot each ordered pair and draw a
smooth curve through the points. Use the graph
to specify the function's domain and range.
SolutionSet up a table of coordinates.
x
0
1
4
9
16
25
( ) 2f x x ( , ) or ( , ( )x y x f x
(0) 0 2 2f (0, 2)
(1) 1 2 3f
(4) 4 2 4f
(9) 9 2 5f
(16) 16 2 6f
(25) 25 2 7f
(1,3)
(4,4)
(9,5)
(16,6)
(25,7)
Solution Contd.
Now we plot the six points and draw a smoth
curve through them. The points on the graph
of have -coordinates that start at zero and
extend indefinitely far to the right. Thus, the
domain consists of al
f x
l real numbers greater
than or equal to 0, represented by [0, ).
Solution Contd.
Graph of ( ) 2f x x
Solution Contd.
Similarly, the points on the graph have
-coordinates that start at 2 and extend
indefinitely upward. Thus, the range
consists of all real numbers greater
than or equal to 2, represented by [2, ).
y
Solution Contd.Graph of ( ) 2f x x
Practice Exercise
Graph the function ( ) | | 1. Use the
integer values 3, 2, 1,0,1,2,3 to
obtain ordered pairs. Use the graph to
specify the function's domain and range.
f x x
x
AnswerGraph of ( ) | | 1f x x
Domain:
Range:
( , )
[1, )
Obtaining Information From Graphs
You can obtain information about a function from its graph. At the right or left of a graph, you will find closed dots, open dots, or arrows.
An arrow indicates that the graph extends indefinitely in the direction in which the arrow points.
Obtaining Information From Graphs A closed dot indicates that the
graph does not extend beyond this point and the point belongs to the graph.
An open dot indicates that the graph does not extend beyond this point and the point does not belong to the graph.
Ex 2: Obtaining Information From a Function’s Graph
Use the graph of the function to answer
the following questions.
a. What are the function values ( 1),
(0) and (1)?
b. What is the domain of ?
c. What is the range of ?
f
f
f f
f
f
Solution of part a.
Because ( 1,4), (0,3), and (1,0)
are points on the graph of ,
the -coordinate, 4 is the ( 1),
the -coordinate, 3 is the (0), and
the -coordinate, 0 is the (1).
f
y f
y f
y f
Thus, ( 1) 4, (0) 3, (1) 0.f f f
Solution of part b.The arrow on the left shows that the
graph extends indefinitely to left.
The close dot on the right shows that
2 is in the domain of and does
not extend beyond this point. Thus, the
domain of is {
x f f
f x
| 2} or the interval
( , 2].
x
Solution of part c.
The points on the graph all have
-coordinates from to 4,
including 4. The graph does not
extends above 4.
Thus, the range of is { | 4}
or the interval ( , 4].
y
y
f y x
The Vertical Line Test
If any vertical line intersects a graph in more than one point, the graph does not define y as a function of x.
Ex 3: Using the Vertical Line Test
Use the vertical line test to identify graphs
in which is a function of .y x
a. b.
Solution
is a function of .y x
a.b.
is not a function of .
Two values of
correspond to an -value.
y x
y
x
Increasing Function
A function isincreasing on aninterval if for any x1
and x2 in the interval,
where x1<x2, then
f(x1)<f(x2).
1 1( , ( ))x f x
2 2( , ( ))x f x
1x 2x
Decreasing Function
A function isDecreasing on aninterval if for any x1
and x2 in the interval,
where x1<x2, then
f(x1)>f(x2).
1 1( , ( ))x f x
2 2( , ( ))x f x
1x 2x
Constant Function
A function isconstant on aninterval if for any x1
and x2 in the interval,
where x1<x2, then
f(x1)=f(x2).
1 1( , ( ))x f x
2 2( , ( ))x f x
1x 2x
Ex 4: Intervals on Which a Function Increases, Decreases, or Is Constant
Describe the increasing, decreasing, or
constant behavior of the function whose
graph is shown.
SolutionAlthough the function's equations are not
given, the graph indicates that the function
is defined in four pieces.
The function is constant on the intervals
( , 4) and (2, ).
The function is decreasing on
the interval
( 4,2).
Even Function
The function is an if
for all in the domain of .
The right side of the equation of an even
function does not change
even functi
if is replaced
with
(
.
on
) ( )
f
xf x
x
f
x
x f
Odd Function
The function is an if
for all in the domain of .
Every term in the right side of the equation
of an odd function changes sign
odd functio
if is
re
n
( ) (
placed .
)
with
f
x f
x
x x
x
f f
Identifying Even or Odd Functions
3
2
2 4
Identify each of the following functions
as even, odd, or neither.
a. ( )
b. ( )
c. ( ) 2
f x x x
g x x x
h x x x
Solution
In each case, replace with and simplify.
If the right side of the equation stays the same,
the function is even. If every term on the right
changes sign, the function is odd.
x x
Solution for part a.
3 ( ) Given func ion tx x xf
3( )f x x x
Replace with .x x 3( )x x xf
3( )f x x x
( ) ( )f x f x Factor
Simplify.
3Replace with ( ).x x f x
Thus, is an odd function.f
Solution for part b.
2 ( ) Given func ion tx x xg
2( )g x x x
Replace with .x x 2( )x x xg
Thus, ( ) ( ) or ( ) ( ).g x g x g x g x Simplify.
We conclude that is neither an even
nor an odd function.
g
Solution for part c.2 4 ( ) 2 Given f unctio nx x xh
2 4( ) 2h x x x
Replace with .x x 2 4( 2)x xh x
( ) ( )h x h x
Simplify.
2 4Replace 2 with h( ).x x x
Thus, is an even function.h
Even Functions and y-Axis Symmetry
The graph of an even function in which
( ) ( ) is symmetric with respect
to the -axis.
f x f x
y
Odd Functions and Origin Symmetry
The graph of an odd function in which
( ) ( ) is symmetric with respect
to the origin.
f x f x
Graphs of Common Functions
Use a graphing utility to verify the following six graphs.
Constant Function Domain: Range: the single
number Constant on
Even function
( )f x c
(,)
c
(,)
Identity Function( )f x x
Domain:Range:Increasing on
Odd function
( , )
( , )
( , )
Standard Quadratic Function
Domain:
Range:
Decreasing on and
increasing on
Even function
( , )
[0, )( ,0) (0, )
2( )f x x
Standard Cubic Function
3( )f x x
Domain:
Range:
Increasing on
Odd function
( , )
( , )
( , )
Square Root Function
( )f x x
Domain:
Range:
Increasing on
Neither even nor odd
[0, )[0, )
[0, )
Absolute Value Function
( ) | |f x x
Domain:Range:
Decreasing on and
incerasing on
Even function
( , ) ( , )
( ,0) (0, )
Greatest Integer Function
( ) int( )
Greatest integer less than or equal to
f x x x
x
The graph of the
greatest integer
function ( ) int( ).f x x
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