Inverse function

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An inverse function is a concept of mathematics. A function will calculate some output ${\displaystyle y}$, given some input ${\displaystyle x}$. This is usually written ${\displaystyle f(x)=y}$. The inverse function does the reverse. Let's say ${\displaystyle g()}$ is the inverse function of ${\displaystyle f()}$, then ${\displaystyle g(y)=x}$. Or otherwise put, ${\displaystyle f(g(x))=x}$. An inverse function to ${\displaystyle f(\ldots )}$ is usually called ${\displaystyle f^{-1}(\ldots ).}$ Do not confuse ${\displaystyle f^{-1}(\ldots )}$ with ${\displaystyle f(\ldots )^{-1}}$: 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 ${\displaystyle f(x)=x^{3}}$ over real ${\displaystyle x}$. Then, ${\displaystyle f^{-1}(x)={\sqrt[{3}]{x}}.}$

At first, make ${\displaystyle y=f(x)}$to ${\displaystyle x=f(y)}$. For example, ${\displaystyle y=e^{x}}$ to ${\displaystyle x=e^{y}}$, It also ${\displaystyle \ln x=y}$, so It's inverse function is ${\displaystyle y=\ln x}$.

Not all functions have inverse functions: for example, function ${\displaystyle f(x)=|x|}$ has none (because ${\displaystyle |-1|=1=|1|}$, and ${\displaystyle f^{-1}(x)}$ 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.