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.


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