Inverse of a function

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

For example, consider the function $f$ given by the rule $\displaystyle f(x) = x^3 + 6$. The function $g(x) = \sqrt[3]{x-6}$ 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. For example, in our example above, $g$ is both a right and left inverse to $f$ on the real numbers.

Often the inverse of a function $f$ is denoted by $f^{-1}$. Note that the $-1$ does not indicate an exponent.

The inverse of a function $f: A \to B$ with range $C$ is a function $f^{-1}: C \to A$ if and only if $f$ is injective, so that every element in the range is mapped from a distinct element in the domain. When the domain and range are subsets of the real numbers, one way to test this is the horizontal line test: if every horizontal line drawn in the plane intersects the graph of the function in at most one point, the function is injective.

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