If two functions are mirror images of each other about the line $y=x$, are they inverses of each other?

Yes, because if one function is $y = f(x)$, the other is $x = g(y)$, since swapping $x$ and $y$ has the same effect as reflecting across the line $y=x$.

Substituting the first into the second, $x = g(y) = g(f(x))$, or in other words, $g^{-1} (x) = f(x)$ if $g^{-1} (x)$ exists. Similarly, substituting the other way around, $y = f(x) = f(g(y)$, so $f^{-1} (y) = g(y)$ if $f^{-1} (y)$ exists.