Question

f(g(x)) = x^4 + 1/x^4 e g(x) = x^2 + 1/x^2, qual é o valor de f(5)?

Answer 1

Explicação passo-a-passo:

Veja que:

$\sf \left(x^2+\dfrac{1}{x^2}\right)^2=x^4+2\cdot x^2\cdot\dfrac{1}{x^2}+\dfrac{1}{x^4}$

$\sf \left(x^2+\dfrac{1}{x^2}\right)^2=x^4+\dfrac{1}{x^4}+2$

Logo, $\sf f(x)=x^2-2$, pois:

$\sf \underbrace{\left(x^2+\dfrac{1}{x^2}\right)^2}_{[g(x)]^2}-2=x^4+\dfrac{1}{x^4}+2-2$

$\sf \left(x^2+\dfrac{1}{x^2}\right)^2=\underbrace{x^4+\dfrac{1}{x^4}}_{f(g(x))}$

Assim:

$\sf f(5)=5^2-2$

$\sf f(5)=25-2$

$\sf f(5)=23$