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I think the easiest way is to calculate g(f(x)) or f(g(x)), but I don't know if it works in every case.

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That's the definition, It does work when you can actually calculate :). A way to get a ballpark is to see if the graphs of $f$ and $g$ are a reflection of each other around the line $x=y$. – gt6989b Nov 29 '12 at 16:25
You could also try taking the equation $y = f(x)$ and solving it for $x$, which will end up giving you $g(y) = x$ if $g$ is the inverse of $f$. – Brad Nov 29 '12 at 16:27

To verify or prove that $g(x)$ is the inverse of $f(x)$, you need to show that $g(f(x)) = f(g(x)) = x$.

That is, you show both $g(f(x)) = x$ and $f(g(x)) = x$.

If you can show this, then you can conclude $g(x) = f^{-1}(x)$. (Alternatively, and equivalently, you can conclude $f(x) = g^{-1}(x)$).

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If f(g(x)) = x and g(f(x)) = x, then you can say that f(x) is the g(x) inverse function.

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