ch 4 - logarithmic and exponential functions - overview n 4.1 - inverse functions n 4.2 -...
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Ch 4 - Logarithmic and Exponential Functions - Overview
4.1 - Inverse Functions 4.2 - Logarithmic and Exponential Functions 4.3 - Derivatives of Logarithmic and
Exponential Functions 4.4 - Derivatives of Inverse Trigonometric
Functions 4.5 - L’Hopital’s Rule; Indeterminate Forms
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4.1 - Inverse Functions(page 242-250)
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Steps For Finding a Functions Inverse
1. Change f(x) to y 2. Switch x and y 3. Solve for y 4. Replace y with 1f x
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Example 3(page 244)
3 2
3 2
f x x
y x
3 2x y 2 3 2x y
212
3x y
1 212
3f x x
2Domain :
3
Range : 0
D x x
R f x f x
1
Domain : 0
2Range :
3
D x x
R f x f x
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Determining Whether Two Functions are Inverses
Two functions are inverses if the meet the following definition.
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Determining Whether Two Functions are Inverses - Example
3 2f x x
212
3g x x
Determine whether f and g are inverse functions
g f x
213 2 2
3x
x2
Domain :3
D x x
f g x
213 2 2
3x
Domain : 0D x x
x
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Horizontal Line Test(page 245)
The Horizontal Line Test is used to determine whether a function would have an inverse over its natural domain.
If a horizontal line is drawn anywhere through the graph of a function and the horizontal line does not intersect the graph in more that one point, then the function passes the horizontal line test.
When a function passes the horizontal line test, the function referred to as one-to-one function. The function is also said to be invertible.
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Horizontal Line Test(page 245)
Functions not passing the horizontal line test must have theirdomains restricted in order to work with their inverses.
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Graphs of Inverse Functions(page 246)
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Graphs of Inverse Functions(page 246)
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Graphs of Inverse Functions(page 246)
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Graphs of Inverse Functions(page 246)
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Increasing or Decreasing Functions Have Inverses
(page 246) If the graph of a function f is always increasing or
always decreasing over the domain of f, then the function f has an inverse over its entire natural domain.
The derivative of a function (slopes of the tangent lines) determines whether a function is increasing or decreasing over an interval.
So, the following theorem suggest that we can determine whether or not a function has an inverse over its entire domain (passes the horizontal line test).
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Example 8(page 247)
for all x.
4However, there is now easy way to solve 5 1, if you switch
and and solve for .
y x
x y y
So, even though we know that f has an inverse, we can notProduce a formula for it.
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Restricting the Domain to Make Functions Invertible
(page 247)
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Continuity and Differentiability of Inverse Functions
(page 248)
If a function is differentiable over an interval, then it is continuous over that interval.
If a function is continuous over an interval, it is notnecessarily differentiable. ( Corner point, Point of vertical tangency, or Point of discontinuity.