# Inverse function

Consider the sets A = {0,2,4,6,8} and B = {1,3,5,7,9} and the function f: A B is defined by y = x + 1. The function f is represented in the diagram below: The function f is bijector. Each x element of A is associated with a single y element of B, so that y = x + 1. However, since f is a bijector, each element y of B is associated with a single element x of A, so that x = y-1; so we have another function g: B A, so that x = y-1 or g (y) = y-1. This function is represented in the diagram below: From what we have just seen, function f takes x to y, while function g takes y to x. The g: B function A is named after inverse function of f and is indicated by f-1.

The domain of f is the image set of g, and the image set of f is the domain of g. When we want, from the sentence y = f (x), to get the sentence of f-1(x), we must perform the following steps:

1º) We isolate x in the sentence y = f (x)
2) Because it is usual the letter x as a symbol of the independent variable, we exchange x for y and y for x.

For example, to get the inverse function of f: IR IR defined by y = 2x + 1, we must:

1) isolate x at y = 2x + 1. So y = 2x + 1 y-1 = 2x x = (y-1) / 2
2) exchange x for y and y for x: y = (x-1) / 2.

So the inverse function of f is: f-1(x) = (x-1) / 2.

Note: For a function f to admit the inverse f-1 she needs to be a bijetora. If f is not a bijet, it has no inverse.

Exercise solved * This content was created by Just Mathematics. The graphs and diagrams were taken from the book Mathematics - Single Volume. Ed. Hail.

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