Question

as funções f:R→R e g:R→R sao definidas por f(x)=2x+3 e g(x)=3x+m. sabendo que f(g(x)) = g(f(x)), determine f(m)

Answer 1

Vamos começar achando f(g(x)):

$f(g(x))~=~2~.~(3x+m)~+~3\\\\\\f(g(x))~=~2~.~3x~+~2~.~m~+~3\\\\\\\boxed{f(g(x))~=~6x+2m+3}$

Vamos agora achar g(f(x)):

$g(f(x))~=~3~.~(2x+3)~+~m\\\\\\g(f(x))~=~3~.~2x~+~3~.~3~+~m\\\\\\\boxed{g(f(x))~=~6x+m+9}$

Como sugere o enunciado, vamos igualar f(g(x)) = g(f(x)):

$6x+2m+3~=~6x+m+9\\\\\\6x-6x+2m-m~=~9-3\\\\\\\boxed{m~=~6}$

Por fim, podemos achar f(m) pedido:

$f(m)~=~f(6)~=~2~.~(6)~+~3~=~12+3~=~\boxed{15}$

Resposta: f(m) = 15