# Inverse Functions

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If f^{-1} is the inverse of function f then (f^{-1} o f)=x or f^{-1}(f(x))=x.

Example: Prove g(x)=5/(x-1) is an inverse of f(x)=1+(5/x)

f(g(x))=1+(5/(5/(x-1)))

f(g(x))=1+(x-1)

f(g(x))=1+x-1=x

Example: Find the inverse function of f given by the following,

f(x) = (x-3)^{2} for x>=3 (this has to hold so that the inverse function does exist)

The function is written in the following format,

y = (x - 3)^{2}

As a result, there are two possible solutions given below:

x - 3 = sqrt(y) or x - 3 = - sqrt(y)

However since sqrt(y) is always larger than zero and x >= 3, the first solution is chosen,

x - 3 = sqrt(y), therefore, x = sqrt(y) + 3;

f^{-1} (y) = sqrt(y) + 3 or f^{-1}(x) = sqrt(x) + 3;

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