 Solution

Seek an affine function $f(x)=cx+d$. Its function composition with itself isEquating coefficients requires $c^2=a$ and $(c+1)d=b$. Because $a>0$ and $a\ne 1$, either choice $c=\sqrt{a}$ or $c=-\sqrt{a}$ has $c+1\ne 0$. Thus two valid functions areSubstitution into $c^{2}x+(c+1)d$ verifies each answer. These provide the requested maps; no assumption that every possible solution is affine function is needed.