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.

Related pages

• Inverse element
• Inverse triangle function
• Inverse hyperbolic function
• Invertible matrix
• Reciprocal