Inverse Functions

Inverse Functions - f -1 is continuous on its domain. 2. If...

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Inverse Functions Inverse Functions Section 5.3
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Reflective Property of Reflective Property of Inverse Functions Inverse Functions The graph of f contains the point (a, b) iff the graph of f -1 contains the point (b, a)
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The Existence of an Inverse The Existence of an Inverse Function Function 1. A function has an inverse iff it is one-to-one. 2. If f is strictly monotonic on its entire domain, then it is one-to-one and therefore has an inverse. Use horizontal line test to determine if a function has an inverse.
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Guidelines for Finding the Inverse Guidelines for Finding the Inverse of a Function of a Function 1. Determine whether the function has an inverse. 2. Solve for x as a function of y. 3. Interchange x and y. 4. Define the domain of f -1 to be the range of f . 5. Verify that f(f -1 (x))= x and f -1 (f(x))=x
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Continuity and Differentiability of Continuity and Differentiability of Inverse Functions Inverse Functions 1. If f is continuous on its domain, then
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Unformatted text preview: f -1 is continuous on its domain. 2. If f is increasing on its domain, then f -1 is increasing on its domain. 3. If f is decreasing on its domain, then f -1 is decreasing on its domain. 4. If f is differentiable at c and f -1 (c) = 0, then f -1 is differentiable at f(c). The Derivative of an Inverse The Derivative of an Inverse Function Function Let f be a function that is differentiable on an interval I. If f has an inverse function g , then g is differentiable at any x for which f(g(x)) = 0. )) ( ( ' , )) ( ( ' 1 ) ( ' = x g f x g f x g Graphs of inverse functions have reciprocal slopes at points (a, b) and (b, a). Look at Figure 5.16 on pg. 334 Look at Example 6 pg. 334. Joke Time Joke Time What do cats eat for breakfast? Mice Krispies What did the canary say when its new cage fell apart? CHEEP CHEEP Where does a sheep go to get a hair cut? To the baa baa shop!...
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Inverse Functions - f -1 is continuous on its domain. 2. If...

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