Inverse Functions

Inverse function is a function that undoes another function: If an input $x$ into the function $f$ produces an output $y$, then putting $y$ into the inverse function $g$ produces the output $x$, and vice versa. i.e., $f(x)=y$, and $g(y)=x$. More directly, $g(f(x))=x$, meaning $g(x)$ composed with $f(x)$ leaves $x$ unchanged.

A function $f$ that has an inverse is called invertible; the inverse function is then uniquely determined by $f$ and is denoted by $f^{-1}$ (read f inverse, not to be confused with exponentiation). If $f$ is invertible, the function $g$ is unique; in other words, there is exactly one function $g$ satisfying this property (no more, no less). Not all functions have an inverse. For this rule to be applicable, for a function whose domain is the set $X$ and whose range is the set $Y$, each element $y \in Y$ must correspond to no more than one $x \in X$; a function $f$ with this property is called one-to-one, or information-preserving, or an injection.

A Function and its Inverse

A function $f$ and its inverse $f^{-1}$. Because $f$ maps $a$ to 3, the inverse $f^{-1}$ maps 3 back to $a$.

Examples

Inverse operations that lead to inverse functions

Inverse operations are the opposite of direct variation functions. Direct variation function are based on multiplication; $y=kx$. The opposite operation of multiplication is division and an inverse variation function is $\displaystyle y=\frac{k}{x}$.

Squaring and square root functions

The function $f(x)=x^2$ may or may not be invertible, depending on the domain. If the domain is the real numbers, then each element in the range $Y$ would correspond to two different elements in the domain $X$ ($\pm x$), and therefore $f$ would not be invertible. More precisely, the square of $x$ is not invertible because it is impossible to deduce from its output the sign of its input. Such a function is called non-injective or information-losing. Notice that neither the square root nor the principal square root function is the inverse of $x^2$ because the first is not single-valued, and the second returns $-x$ when $x$ is negative. If the domain consists of the non-negative numbers, then the function is injective and invertible.