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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 onetoone. If $f$ is not onetoone then the inverse function does not exist unless the domain is restricted.

In contrast, the cubic function $f(x)=x^3$ does have an inverse because it is onetoone and onto.
The problem of functions not having inverses is what motivated the creation of Riemann surfaces.
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DomainOfAFunction Function RiemannSurface SquareRoot 

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