Inverse (functions)

In mathematics, an inverse function (or anti-function) is a function that "reverses" another function: if the function $f$ applied to an input $x$ gives a result of $y$ , then applying its inverse function $g$ to $y$ gives the result $x$ , and vice versa. i.e., $f (x) = y$ if and only if $g (y) = x$ .

As a simple example, consider the real-valued function of a real variable given by $f (x) = 5 x - 7$ . Thinking of this as a step-by-step procedure (namely, take a number $x$ , multiply it by 5, then subtract 7 from the result), to reverse this and get $x$ back from some output value, say $y$ , we should undo each step in reverse order. In this case that means that we should add 7 to $y$ and then divide the result by 5. In functional notation this inverse function would be given by,

With $y = 5 x - 7$ we have that $f (x) = y$ and $g (y) = x$ .

Not all functions have inverse functions. In order for a function $f : X \to Y$ to have an inverse, it must have the property that for every $y$ in $Y$ there must be one, and only one $x$ in $X$ so that $f (x) = y$ . This property ensures that a function $g : Y \to X$ will exist having the necessary relationship with $f$ .

Let $f$ be a function whose domain is the set $X$ , and whose image (range) is the set $Y$ . Then $f$ is invertible if there exists a function $g$ with domain $Y$ and image $X$ , with the property:

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