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Section6.2 - Math 2B Dr C Famiglietti Section 6.2 Inverse...

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Math 2 B Dr. C. Famiglietti ************************************************************************ Section 6.2: Inverse Functions The inverse of the function y = f x ( ) is y = f 1 x ( ) . Comments: f 1 x ( ) 1 f x ( ) since -1 is a superscript and not an exponent. Only a one-to-one function has an inverse (where a one-to-one function is a function that never takes on the same value twice). The Horizontal Line Test is a graphical test which can be used to determine if a function has an inverse (a function with an inverse will pass the Horizontal Line Test). A function that increases throughout its entire domain will have an inverse (since it is one-to-one). Similarly, a function that decreases throughout its entire domain will have an inverse. Therefore, if we cannot use the Horizontal Line Test to show that a function is one-to-one, we can use the Increasing-Decreasing Test to see whether or not a function is one-to-one and has an inverse. The domain of f 1 x ( ) is the range of f x ( ) , and the range of f 1 x ( ) is the domain of f x ( ) . If a , b ( ) is an ordered pair on the graph of f x ( ) , then b , a ( ) is a point on the graph of f 1 x ( ) . In other words, if f a ( ) = b , then f 1 b ( ) = a . The graph of f 1 x ( ) can be obtained by reflecting the graph of f x ( ) about the line y = x . A function and its inverse will satisfy a pair of equations referred to as the cancellation equations: f f 1 ( ) = x and f 1 f ( ) = x . To determine the inverse of a one-to-one function: 1. Write the function as y = f x ( ) . 2. Solve for x in terms of y , if possible. 3. Interchange x and y . (This is done to express the inverse function as a function of x , which is traditionally used as the independent variable.) 4.
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