# Minimum Supremum

Algebra Level 5

$\large f_\alpha(\alpha x ) = f_\alpha(x + \alpha)$

The function $$f_{\alpha} \colon \mathbb{R} \to \mathbb{R}$$ satisfies the above functional equation for all $$x$$ where $$\alpha$$ is real number $$\neq 0, 1$$.

It is given that $$f_{12}(2) = 3$$. Let $$m$$ be the minimum value of the least upper bound of $$k$$ such that $$f_{12}(k) = 3$$.

If $$m$$ can be written as $$\dfrac{A}{B}$$, where $$A$$ and $$B$$ are coprime positive integers, find $$A+B$$.

Bonus: Find all functions $$f_{\alpha}(x)$$ that satisfy the given functional equation.

×