If $y=f(x)$ is a function, the inverse function $g$ is the function such that $x=g(f(x)).$ In a sense, the function $g$ "undoes" the action of $f.$

For a general function $f,$ the inverse function only exists if $f$ is one-to-one. If $f$ is not one-to-one then the inverse function does not exist unless the domain is restricted.

For example, if $f(x)=x^2,$ the inverse function does not exist unless we restrict the domain to just the non-negative numbers, and one of the square roots is chosen.

In contrast, the cubic function $f(x)=x^3$ does have an inverse because it is one-to-one and onto.

The problem of functions not having inverses is what motivated the creation of Riemann surfaces.

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