Inverse function

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, ${\displaystyle g(f(x))=x}$. An inverse function to ${\displaystyle f}$ is usually called ${\displaystyle f^{-1}}$.[1] It is not to be confused with ${\displaystyle 1/f}$, which is a reciprocal function.[2]

Examples

If ${\displaystyle f(x)=x^{3}}$ over real ${\displaystyle x}$, then ${\displaystyle f^{-1}(x)={\sqrt[{3}]{x}}.}$

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

References

1. "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-09-08.
2. Weisstein, Eric W. "Inverse Function". mathworld.wolfram.com. Retrieved 2020-09-08.