I saw a post that said that $f^{-1}(f(x))$ is always equal to $x$. Can anyone explain to me why? I tried googling but the only thing that came close to a proof is this video, but it simply solved the equation.
The equations that made me question this are $f(x) = 3x-2$ and its inverse $f^{-1}(x) = (x+2) / 3$.
Let's say we have a function $f$ such that $y=f(x)$. If $f$ is invertible (has an inverse), this inverse $f^{-1}$ satisfies the property
$$f^{-1}(y)=x$$
We established earlier, however, that $y=f(x)$. This means that
$$f^{-1}(f(x))=x$$
where $x$ is in the domain of $f$.
This is similar to this proof that $f(f^{-1}(x))=x$.