Question

Sejam as funções reais $g(x) = 3x-2$ e $fog(x) = 9 x^{2} - 3x + 1$. Determine a lei da função f.

Answer 1

$fog(x) = f(3x-2)\\\\ f(3x-2) = 9x^2-3x+1\\\\\ 3x-2 = t\\\\ \boxed{x=\frac{t+2}{3}}$

$f(t) = 9[(\frac{t+2}{3})]^2-3(\frac{t+2}{3})+1\\\\ f(t) = 9\frac{(t+2)^2}{9}-(t+2)+1\\\\\ f(t) = (t+2)^2-(t+2)+1\\\\ f(t) = t^2+4t+4-t-2+1\\\\ \boxed{f(t) = t^2+3t+3}$

Portanto:

$\boxed{f(x) = x^2+3x+3}$