Fulfilling conditions of Inverse function(?)

Question

Let's assume that

$$h(g(t))=t$$

What conditions ae needed to say that

$$g(h(t))=t$$

Is satisfied too?

(Provided that $h$ and $g$ are continuous and derivatable, but not knowing whether they have inverse though..)

I am not sure about the title of this question. If you know better title, feel free to let me know or edit this question.

Answer 1

If the domain is $\mathbb R$ then a necessary and sufficient condition is that $h$ is one-to-one. If $h$ is one-to-one change $t$ to $h(t) $ in the hypothesis to get $h(g(h(t))=h(t)$. This implies $g(h(t))=t$ because $h$ is one-to-one. Conversely the equation $g(h(t))=t$ clearly implies that $h$ is one-to-one.