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lecture5 - Lecture 5 Inverse Functions Logarithms(Chapter 1...

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Lecture 5: Inverse Functions, Logarithms (Chapter 1, Sec. 5) Def. A function f is called a one-to-one (1 - 1) function if ex. f ( x ) = x 2 + 1 ex. f ( x ) = x 3 6 - ? 6 - ? Horizontal Line Test A function is one-to-one if and only if any horizontal line intersects the graph of the function in at most one point. NOTE: Increasing/decreasing functions
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Inverse Functions Def. Let f be a one-to-one function with domain A and range B . Then it has a unique inverse function f ¡ 1 : B ! A which assigns to each y in B the unique x value in A given by f ¡ 1 ( y ) = x if and only if Thinking of x as the independent variable, we can also write f ¡ 1 ( x ) = y if and only if Note the following results of the deflnition: 1) If ( x; y ) is a point on the graph of f ( x ), then 2) domain and range of f ¡ 1
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Inverse relationships f ¡ 1 ( f ( x ) ) = for every x in A f ( f ¡ 1 ( x ) ) = for every x in B ex. Show that f ( x ) = x 3 + 1 and g ( x ) = 3 p x ¡ 1 are inverse functions. ex. Find the inverse of f ( x ) = p x + 2. Check domain and range.
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Inverse functions and graphs ex. Given the graph of f ( x ) = x 3 + 1, sketch the graph of the inverse function.
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