Question 5
Let $b$ be a positive real number. Determine all functions $f:\mathbb R\to\mathbb R$ satisfying, for every $x,y\in\mathbb R$,
$$f(x+y)=f(x)3^{b^y+f(y)-1}+b^x\left(3^{b^y+f(y)-1}-b^y\right).$$
$$f(x+y)=f(x)3^{b^y+f(y)-1}+b^x\left(3^{b^y+f(y)-1}-b^y\right).$$