Known Fact: Consider a continuous function $f:\mathbb{R}\rightarrow\mathbb{R}$. Let $g(x)$ be its inverse, i.e. $f(g(x))=g(f(x))=x$. If there is an $x$-axis and $y$-axis and we plot $y=f(x)$, then it is easy to see that its reflection on the line $y=x$ will be the curve $y=g(x)$.
My Question: Now suppose we have functions $f$ and $g$ such that $f(g(x))=g(f(x))=h(x)$ for some differentiable function $h(x)$. My question is now that if we reflect the curve $y=f(x)$ on the curve $y=h(x)$, will the image be $y=g(x)$?