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3.4 Inverse Functions & Relations

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Inverse Relations Two relations are inverses if and only if one relation contains the element (b, a) whenever the other relation contains the element (a, b). Two relations are inverses if and only if one relation contains the element (b, a) whenever the other relation contains the element (a, b). If f(x) denotes a function then f -1 (x) denotes the inverse. The inverse is not necessarily a function. If f(x) denotes a function then f -1 (x) denotes the inverse. The inverse is not necessarily a function. Inverses are symmetric to each other with respect to the line y = x. Inverses are symmetric to each other with respect to the line y = x.

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Ex 1 Graph f(x) = x 2 and its inverse.

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Horizontal line test – used to determine if the inverse of a relation will be a function. Horizontal line test – used to determine if the inverse of a relation will be a function. If every horizontal line intersects the graph of the relation in at most one point, then the inverse of the relation is a function. If every horizontal line intersects the graph of the relation in at most one point, then the inverse of the relation is a function.

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Finding the inverse algebraically – 1. let y = f(x) 2. Interchange x and y 3. Solve the resulting equation for y.

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Ex 2 f(x) = x Is the inverse a function? Is the inverse a function? Find the inverse. Find the inverse. Graph. Graph.

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Ex 3 Graph

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Inverse functions – two functions, f and f -1, are inversed if and only if f(f -1 (x)) = f -1 (f(x)) = x Inverse functions – two functions, f and f -1, are inversed if and only if f(f -1 (x)) = f -1 (f(x)) = x

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Ex 4 Given f(x) = 3x 2 + 7, find f -1 (x) and verify that f and f -1 are inverse functions.

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