Question

Daddos f(x) = 3x + 4 g(x) = 2x +ak, calcule o valor de k para que se tenha (f o g)(x) = (g o f)(x).

Answer 1

$(fog)(x)=f(g(x))\\\\f(g(x))=3(2x+ak)+4\\\\f(g(x))=6x+3ak+4\\\\\boxed{f(g(x))=6x+(3ak+4)}$

 E,

$(gof)(x)=g(f(x))\\\\g(f(x))=2(3x+4)+ak\\\\g(f(x))=6x+8+ak\\\\\boxed{g(f(x))=6x+(ak+8)}$


 Igualando-as,...

$f(g(x))=g(f(x))\\\\6x+(3ak+4)=6x+(ak+8)\\\\6x+3ak+4=6x+ak+8\\\\3ak-ak=6x+8-6x-4\\\\2ak=4\\\\k=\frac{4}{2a}\\\\\boxed{\boxed{k=\frac{2}{a}}}$