Explain how to verify inverse function

Question

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?

Answer 1

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 $f^{-1}(f(x))$.