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How does an inverse function undo the action of an original function?

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A graph of the function f(x) = x cubed minus 3x plus 2 is shown on a blue background, with a red curve passing through key points. Local maximum and minimum points are marked with black dots at x equals negative 1 and x equals 1, illustrating important features of the curve. — function Plot of the cubic equation f(x) = x³ − 3x + 2. The plotted points are where changes in curvature occur.

An inverse function undoes the effect of an original function by interchanging the roles of the independent and dependent variables. If a function $f$ maps an input $x$ to an output $y$ (expressed as $y = f (x)$ ), its inverse, denoted as $f^{- 1}$ , maps $y$ back to the original input $x$ .1

Mathematically, this relationship is defined such that applying the inverse function to the result of the original function returns the variable to its starting state. Specifically, for a function $f$ and its inverse $g$ (or $f^{- 1}$ ), the following conditions must hold:
* $g (f (x)) = x$
* $f (g (y)) = y$

By reversing the mapping, inverse functions allow for the resolution of equations where the original operation needs to be neutralized. For instance, if a function doubles a value ( $f (x) = 2 x$ ), its inverse divides that value by two ( $f^{- 1} (x) = x / 2$ ), effectively returning the input to its original form.1