Question

Dada as funçoes f(x)= √x-5 (os dois na raiz quadrada) e g(x)= x+4, determina f o g(r), g o f(r) e suas inversas

Answer 1

Para calcular a fog(x), a função g(x) será considerada variável da função f(x), acompanhe:

$fog(x)~=~\sqrt{g(x)~-~5}\\\\\\fog(x)~=~\sqrt{(x+4)~-~5}\\\\\\\boxed{fog(x)~=~\sqrt{x-1}}$

Para calcular a gof(x), a função f(x) será considerada variável da função g(x), acompanhe:

$gof(x)~=~f(x)~+~4\\\\\\\boxed{gof(x)~=~\sqrt{x-5}~+~4}$

Para determinar as inversas, vamos seguir 2 passos:

--> Trocar "x" por "y";

--> Isolar "y".

Vamos determinar a inversa da fog(x):

$fog(x)~=~y\\\\\\y~=~\sqrt{x-1}\\\\\\Trocando~"x"~por~"y"\\\\\\x~=~\sqrt{y-1}\\\\\\Isolando~"y"\\\\\\x^2~=~y-1\\\\\\\boxed{y~=~fog^{-1}(x)~=~x^2+1}$

Vamos determinar a inversa da gof(x):

$gof(x)~=~y\\\\\\y~=~\sqrt{x-5}~+~4\\\\\\Trocando~"x"~por~"y"\\\\\\x~=~\sqrt{y-5}~+~4\\\\\\Isolando~"y"\\\\\\x-4~=~\sqrt{y-5}\\\\\\(x-4)^2~=~y-5\\\\\\\boxed{y~=~gof^{-1}(x)~=~(x-4)^2+5}$