Functions satisfying $Af(x) + B = f(Cx)$

Question

Let $f: [0,\infty) \to [0,\infty)$ be a continuous function such that $Af(x) + B = f(Cx)$ for all $x \in [0,\infty)$. Here the constants satisfies $A \in (0,1), B>0, C>1$.

Can we find $f$, or even it is unique?

Answer 1

Hint: try common function patterns. One you might try is a power function $kx^\alpha$. Except the presence of the $B$ here might make you try $f(x)=kx^\alpha+b$.

When you try such a function in your relation, can you work out what things like $k,\alpha,b$ would have to be in terms of $A,B,C$?