Inverse of a function

The inverse of a function is a function that "undoes" the action of a given function.

For example, consider the function $f: \mathbb R \to R$ given by the rule $\displaystyle f(x) = x^2 + 6$ . The function $g(x) = \sqrt{x-6}$ has the property that $f(g(x)) = x$ . In this case, $g$ is called the (right) inverse function of $f$ . (Similarly, a function $g$ such that $g(f(x))=x$ is called the left inverse function of $f$ . Typically the right and left inverses coincide on a suitable domain, and in this case we simply call the right and left inverse function the inverse function. Often the inverse of a function $f$ is denoted by $f^{-1}$ (the $-1$ does not indicate a exponent).

The inverse of a function is only a function itself iff the original function is one-to-one, or that every element in the range is paired with a distinct element in the domain. A way to test this is the horizontal line test, where if a horizontal line can be drawn through the graph of a function and touch two points on the graph, the function is not one-to-one.

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Inverse of a function