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Inverse Functions solutions

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Inverse Functions solutions
February 11, 2019
Q:\1. Curriculum Subjects\Mathematics\Graphing Package
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1
Inverse Functions solutions
February 11, 2019
Fact:
If the function
has an inverse function then this inverse function:
• is denoted
Note:
is not the reciprocal of
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i.e.
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Inverse Functions solutions
February 11, 2019
Fact:
If the function
has an inverse function then this inverse function:
• is the reflection of
Consider
in the line
f(x)
y=x
(Note: broken line)
f ­1( x )
Reflected
point
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3
Inverse Functions solutions
February 11, 2019
Fact:
If the function
• must be a function
has an inverse function then this inverse function:
i.e. must satisfy the vertical line test
Consider
vertical line test
f(x)
f ­1( x )
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Inverse Functions solutions
February 11, 2019
Consider
vertical line test
f ( x ) is a function
but
f ­1( x ) is not a function
Thus, no inverse function
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5
Inverse Functions solutions
February 11, 2019
Consider
vertical line test
Both f ( x ) and f ­1( x )
are functions
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6
Inverse Functions solutions
February 11, 2019
Consider
vertical line test
f ( x ) is a function
but
f ­1( x ) is not a function
Thus, there is no inverse function
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7
Inverse Functions solutions
February 11, 2019
Fact:
If the function
has an inverse function then this inverse function:
• can be pre­determined using the horizontal line test
Consider
horizontal line test
f(x)
f ( x ) will have an
inverse function as it
passes the horizontal
line test
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Inverse Functions solutions
February 11, 2019
horizontal line test
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Inverse Functions solutions
February 11, 2019
Consider functions with a restricted domain
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Inverse Functions solutions
February 11, 2019
Consider functions with a restricted domain
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Inverse Functions solutions
February 11, 2019
Features of a function and its inverse function
Consider again the function
1. Features of f(x) and f ­1(x)
Features
f(x)
f ­1(x)
Slope
x­intercept
y­intercept
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Inverse Functions solutions
Features
f(x)
February 11, 2019
f ­1(x)
Asymptote
x­intercept
y­intercept
Domain
Range
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Inverse Functions solutions
February 11, 2019
Consider again the function
2. Find the equation of f ­1(x) using coordinate geometry
Features
f(x)
Slope
2
f ­1(x)
y­intercept
x­intercept
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Inverse Functions solutions
February 11, 2019
Consider again the function
2. Find the equation of f ­1(x) using variable interchange
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Inverse Functions solutions
February 11, 2019
Fact:
If the function
has an inverse function then this inverse function:
• has the property
• where
3. Check that
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is the identity function
OR
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Inverse Functions solutions
February 11, 2019
Consider the reciprocal function
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Inverse Functions solutions
February 11, 2019
Summary of facts
If the function
has an inverse function then this inverse function:
• is denoted
• is the reflection of
• must be a function
in the line
i.e. must satisfy the vertical line test
• can be pre­determined using the horizontal line test
• has the property
• where
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is the identity function
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Inverse Functions solutions
Consider
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. Find
February 11, 2019
using variable interchange.
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Inverse Functions solutions
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February 11, 2019
20
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