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Graphs of Functions
Lesson 3
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Warm Up – Perform the Operations and Simplify
2
412.
x
a
5420
3.
22
xx
x
xxb
352
10
5
6112.
2
23
xx
x
x
xxxc
92
7
92
7.
x
x
x
xd
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2
412.
x
a
Solution
2
20122
424122
4
2
2122
412
x
xx
xxx
xx
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Solution
5420
3.
22
xx
x
xxb
145
3
145
433
145
413
1545
3
2
2
xxx
xx
xxx
xxx
xxx
xxx
xx
x
xx
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352
10
5
6112.
2
23
xx
x
x
xxxc
Solution
35
106
3125
10612
312
10
5
6112
352
10
5
6112
2
2
23
x
xx
xxx
xxxx
xx
x
x
xxx
xx
x
x
xxx
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92
7
92
7.
x
x
x
xd
Solution
7
7
7
92
92
7
92
7
92
7
x
x
x
x
x
x
x
x
x
x
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Domain & Range of a Function
What is the domain ofthe graph of the functionf?
4,1: A
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Domain & Range of a Function
What is the range ofthe graph of the functionf?
4,5
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Domain & Range of a Function .21 fandfFind
51 f
42 f
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Let’s look at domain and range of a function using an algebraic approach.
Then, let’s check it with a graphical approach.
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Find the domain and range of
Algebraic Approach
.4 xxf
The expression under the radical can not be negative.Therefore, Domain .04 x
,4
4:
or
xA Since the domain is never negative the range is the set of all nonnegative real numbers.
,0
0:
or
yARange
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Find the domain and range of
Graphical Approach
.4 xxf
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Increasing and Decreasing Functions
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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.
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Falls from x = -2 to x = 0.
Is constant from x = 0 to x = 2.
Rises from x = 2 to x = 4.
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Ex: Find the open intervals on which the function is increasing, decreasing, or constant.
Increases over the entire real line.
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Ex: Find the open intervals on which the function is increasing, decreasing, or constant.
,11,
:
and
INCREASING
1,1
:
DECREASING
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Ex: Find the open intervals on which the function is increasing, decreasing, or constant.
0,
:
INCREASING
2,0
:CONSTANT
.2
:DECREASING
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Relative Minimum and Maximum Values
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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.
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General Points – We’ll find EXACT points later……
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Approximating a Relative Minimum
Example: Use a GDC to approximate the relative minimum of the function given by
.243 2 xxxf
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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.
.243 2 xxxf
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Example
Use a GDC to approximate the relative minimum and relative maximum of the function given by
.3 xxxf
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Solution
Relative Minimum
(-0.58, -0.38)
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Solution
Relative Maximum
(0.58, 0.38)
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Step Functions and Piecewise-Defined Functions
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Because of the vertical jumps, the greatest integer function is an example
of a step function.
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Let’s graph a Piecewise-Defined Function
Sketch the graph of
1,4
1,32
xx
xxxf
Notice when open dots and closeddots are used. Why?
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Even and Odd Functions
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Graphically
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Algebraically
Let’s look at the graphs again and see if this applies.
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Graphically
☺ ☺
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Example
Determine whether each function is even, odd, or neither.
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AlgebraicGraphical – Symmetric to Origin
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Algebraic
Graphical – Symmetric to y-axis
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Algebraic
Graphical – NOT Symmetric to origin OR y-axis.
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You Try
Is the function
Even, Odd, of Neither?
xxf
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Solution xxf
Symmetric about the y-axis.