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Let $f(x) = 3x -1$

Can someone explain how to verify $f[f^{-1}(x)] = x$ and $f^{-1}[f(x)] = x$, each for $x$ in the appropriate domain?

I was able to determine that the inverse function of $f(x) = 3x - 1$ is $\frac {x + 1}{3}$.

Do I just substitute in the known equations and solve?

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    No problem; it's kinda important here as feedback, you see. The check mark is our way of knowing that you were satisfied by the answer you have marked.2011-10-25

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You just write the "instructions" $f(f^{-1}(x))$ and $f^{-1}(f(x))$. That is,

$ f(f^{-1}(x)) = f\left( \frac{x+1}{3} \right) = 3\frac{x+1}{3} -1 = x + 1 -1 = x \ . $

And similarly for $f^{-1}(f(x))$.