Question

Se fog= 48 x² - 24 x + 8 e g(x)= 4x -1,determine a expressão de f(x)

Answer 1

$fog(x)=48x^{2}-24x+8\\\\f(g(x))=48x^{2}-24x+8\\\\f(4x-1)=48x^{2}-24x+8\\\\f(4k-1)=48k^{2}-24k+8$

Queremos encontrar f(x). Ou seja, queremos encontrar um certo k que faça

$4k-1=x\\\\4k=x+1\\\\\boxed{\boxed{k=\dfrac{x+1}{4}}}$

Portanto, se k = (x + 1) / 4, f(4k - 1) = f(x)

Logo, vamos trocar 4k - 1 por x e x por (k + 1) / 4:

$f(4k-1)=f(x)=48\left(\dfrac{x+1}{4}\right)^{2}-24\left(\dfrac{x+1}{4}\right)+8\\\\\\f(x)=48\dfrac{(x+1)^{2}}{4^{2}}-6(x+1)+8\\\\\\f(x)=\dfrac{48}{16}(x+1)^{2}-6(x+1)+8\\\\\\\boxed{\boxed{f(x)=3(x+1)^{2}-6(x+1)+8}}$