Question

se f(x)=-x²+x+1 e (fog)(x)=-x²+3x-1, encontre g(x)

Answer 1

Para facilitar, vamos chamar g(x) de "y". Teremos então a equação:

$f_og(x)~=~-x^2+3x-1\\\\-(y)^2+y+1~=~-x^2+3x-1\\\\\\Para~achar~g(x)~precisaremos~isolar~y\\\\Vamos,~entao, ~completar~quadrado~no~lado~esquerdo~da~equacao:\\\\\\y^2-y-1~=~x^2-3x+1\\\\\\(y-\frac{1}{2})^2-1-\frac{1}{4}~=~x^2-3x+1\\\\\\(y-\frac{1}{2})^2~=~x^2-3x+1+1+\frac{1}{4}\\\\\\(y-\frac{1}{2})^2~=~x^2-3x+\frac{9}{4}\\\\\\(y-\frac{1}{2})^2~=~(x-\frac{3}{2})^2\\\\\\y-\frac{1}{2}~=~\sqrt{(x-\frac{3}{2})^2}\\\\\\y-\frac{1}{2}~=~x-\frac{3}{2}\\\\\\\boxed{y~=~g(x)~=~x-1}$