Expressing functions using compositions

Question

A question I'm doing right now concerns a function $f$, that is defined as $f' \circ h$ where $f'$ and $h$ are functions defined in the question. I was wondering if there's any way of defining $f'$ using $f, h$?
Thanks in advance!

Answer 1

If you know that $f(x) = f(y)$ whenever $h(x) = h(y)$, then there will be such a function $f'$, whose domain will be the image of $h$, and defined by the condition $f'(z) = f(x)$ where $x \in h^{-1}(z) = \{y \mid h(y) = z\}$. The condition means that $f$ is constant on $h^{-1}(z)$, so it does not matter which element you pick.

On the other hand, if there exist even one pair of values $x, y$ with $h(x) = h(y)$ but $f(x) \ne f(y)$, then such a function $f'$ cannot exist. For if $z$ is the common value of $h(x)$ and $h(y)$, then $f(x) = f'\circ h(x) = f'(z)$ and $f(y) = f'\circ h(y) = f'(z)$. Which cannot be, as $f(x) \ne f(y)$.