Section 4-5 Inverse Functions · Inverse Notes •The inverse of a function f is written −1and is...
Transcript of Section 4-5 Inverse Functions · Inverse Notes •The inverse of a function f is written −1and is...
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Section 4-5 Inverse Functions
Objective: To find the inverse of a function, if the inverse
exists.
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FunctionsImagine functions are like the dye you use to color eggs. The white egg (x) is put in the function blue dye, B(x), and the result is a blue egg (y).
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The Inverse Function “undoes” what the function does.
The Inverse Function of the Blue dye is bleach.
The bleach will “undye” the blue egg and make it white.
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In the same way, the inverse of a given function will “undo” what the original function did.
For example, let’s take a look at the square function: f(x) = x2
3
x f(x)
33333 9999999
y 𝒇−𝟏(𝒙)
9999999 3333333
x2 x
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555555 252525
252525
252525255 55555555
In the same way, the inverse of a given function will “undo” what the original function did.
For example, let’s take a look at the square function: f(x) = x2
x f(x) y 𝒇−𝟏(𝒙)
x2 x
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111111111111 121121121121121121121121121121121121121121 1111111111111111
In the same way, the inverse of a given function will “undo” what the original function did.
For example, let’s take a look at the square function: f(x) = x2
x f(x) y 𝒇−𝟏(𝒙)
x2 x
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Inverse Function Definition
Two functions f and g are called inverse functions if the following two statements are true:
1. 𝑔(𝑓 𝑥 ) = 𝑥 for all x in the domain of f.
2. 𝑓(𝑔 𝑥 ) = 𝑥 for all x in the domain of g.
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Graphically, the x and y values of a point are switched.
The point (4, 7)
has an inverse point of (7, 4)
AND
The point (-5, 3)
has an inverse point of (3, -5)
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Graphically, the x and y values of a point are switched.
If the function y = g(x) contains the points
then its inverse, y = g-1(x), contains the points
x 0 1 2 3 4
y 1 2 4 8 16
x 1 2 4 8 16
y 0 1 2 3 4
Where is there a line of reflection?
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The graph of a function and its
inverse are mirror images about the line
𝒚 = 𝒙𝒚 = 𝒇(𝒙)
𝒚 = 𝒇−𝟏(𝒙)
y = x
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Inverse Notes
•The inverse of a function f is written𝑓−1 and is read “f inverse”
• 𝑓−1(𝑥) is read, “f inverse of x”• If point (𝑥, 𝑦) is on graph of f, then point (𝑦, 𝑥) is on the graph of the inverse of f.
• The graph of 𝒇−𝟏is the reflection of the graph of 𝒇 in the line 𝑦 = 𝑥
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Find the inverse of a function algebraically:
Example 1: f(x) = 6x - 12
Step 1: Switch x and y
x = 6y - 12
Step 2: Solve for y
x 6y 12
x 12 6y
x 12
6 y
1
6x 2 y
*Note: You can replace f(x) with y.
𝒇−𝟏 𝒙 =𝟏
𝟔𝒙 + 𝟐
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Given the function: f(x) = 3x2 + 2 Find the inverse.
Step 1: Switch x and yx = 3y2 + 2
Step 2: Solve for yx 3y2 2
x 2 3y2
x 2
3 y2
x 2
3 y
Example 2:
𝒇−𝟏 𝒙 =𝒙 − 𝟐
𝟑
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On the same axes, sketch the graph of
and its inverse.
2,)2(2 xxy
Notice
)0,2(
)1,3(
xy
)4,4(x
Solution:
)2,0(
)3,1(
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On the same axes, sketch the graph of
and its inverse.
2,)2(2 xxy
Noticexy
2)2( xy
Solution:
Using the translation of what is the equation of the inverse function?
x
2 xy
2)(1
xxf
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2)2( xy
2 xy
domain and range
The domain of is . 2x
)(xf
Since is found by swapping x and y,
)(1
xf
2)2()( xxf 2xDomain
2y2)(1
xxf Range
2,)2()(2 xxxfThe previous example used .
the values of the domainof give the values of the range of .
)(xf)(
1xf
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2)2( xy
domain and range
2,)2()(2 xxxfThe previous example used .
The domain of is . 2x
)(xf
2 xySince is found by swapping x and y,
)(1
xf
)(1
xfgive the values of the domain of
the values of the domainof give the values of the range of .
)(xf)(
1xf
Similarly, the values of the range of )(xf
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