Inverse Functions

An 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 $f^{-1}$.

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 $a$.

Instead of considering the inverses for individual inputs and outputs, one can think of the function as sending the whole set of inputs—the domain—to a set of outputs—the range. Let $f$ be a function whose domain is the set $X$ and whose range is the set $Y$. Then $f$ is invertible if there exists a function $g$ with domain $Y$ and range $X$, with the following property:

$f (x) = y \Leftrightarrow g (y) = x$

Inverse Functions

If $f$ maps $X$ to $Y$, then $f^{-1}$  maps $Y$ back to $X$.

If $f$ is invertible, the function $g$ is unique; in other words, there is exactly one function $g$ satisfying this property (no more, no fewer). That function $g$ is then called the inverse of $f$, and is usually denoted as $f^{-1}$.

Stated otherwise, a function is invertible if and only if its inverse relation is a function on the range $Y$, in which case the inverse relation is the inverse function. Not all functions have an inverse. For this rule to be applicable, 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, information-preserving, or an injection.

Example

Let's take the function $y=x^2+2$. To find the inverse of this function, undo each of the operations on the $x$ side of the equation one at a time. We start with the $+2$ operation. Notice that we start in the opposite order of the normal order of operations when we undo operations. The opposite of $+2$ is $-2$. We are left with $x^2$. To undo use the square root operation. Thus, the inverse of $x^2+2$ is $\sqrt{x-2}$. We can check to see if this inverse "undoes" the original function by plugging that function in for $x$:

$\sqrt{\left(x^2+2\right)-2}=\sqrt{x^2}=x$