# Inverse function

An inverse function is a concept of mathematics. A function will calculate some output $y$ , given some input $x$ . This is usually written $f(x)=y$ . The inverse function does the reverse. Let's say $g()$ is the inverse function of $f()$ , then $g(y)=x$ . Or otherwise put, $f(g(x))=x$ . An inverse function to $f(\ldots )$ is usually called $f^{-1}(\ldots ).$ Do not confuse $f^{-1}(\ldots )$ with $f(\ldots )^{-1}$ : the first is a value of an inverse function, the second is reciprocal of a value of a normal function.

## Examples

Let's take a function $f(x)=x^{3}$ over real $x$ . Then, $f^{-1}(x)={\sqrt[{3}]{x}}.$ At first, make $y=f(x)$ to $x=f(y)$ . For example, $y=e^{x}$ to $x=e^{y}$ , It also $\ln x=y$ , so It's inverse function is $y=\ln x$ .

Not all functions have inverse functions: for example, function $f(x)=|x|$ has none (because $|-1|=1=|1|$ , and $f^{-1}(x)$ should give both 1 and -1 when given 1)), but every binary relation has its own inverse relation.

Finding the inverse of a function can be very difficult to do.