# 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, $g(f(x))=x$ . An inverse function to $f$ is usually called $f^{-1}$ . It is not to be confused with $1/f$ , which is a reciprocal function.

## Examples

If $f(x)=x^{3}$ over real $x$ , then $f^{-1}(x)={\sqrt[{3}]{x}}.$ To find the inverse function, swap the roles of $x$ and $y$ and solve for $y$ . For example, $y=e^{x}$ would turn to $x=e^{y}$ , and then $\ln x=y$ . This shows that the inverse function of $y=e^{x}$ is $y=\ln x$ .

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

In some cases, finding the inverse of a function can be very difficult to do.