function equation with translation of independent variable: $\frac{f(x+a)}{f(a)}=g(x)$

Question

The following has come up in some work I'm doing: If $\frac{f(x+a)}{f(a)}=g(x)$, where $g(x)$ is given and $a \ge 0$ is a constant, what is $f(x)$? We can assume that $g(x)>0 \ \forall x$ . Of course a solution would be great, but I'd appreciate even general information on this equation, such as how it would be referred to (functional equation with translation?), similar equations, etc. Thank you.

Answer 1

$f(x) = e^x$ and $g(x) = e^x$, in fact: $$f(x+a) = e^{x+a} = e^x e^{a} = f(a) g(x)$$

Note that $g(x) = e^x > 0 ~ \forall x$ as for hypotesis.

Answer 2

We have $f(x)=f(x-a+a)=g(x-a)f(a)$. Especially for $x=a$ we have $f(a)=g(0)f(a)$ and as $f(a)\neq 0$ we must have $g(0)=1$. This means there exists a solution only if $g(0)=1$ and in that case it is $f(x)=g(x-a)c$ where the constant $f(a)=c$ can be chosen.