Question

Sejam f e g funções reais tais que f(x) = 3ˣ e g(x) = x - 2. Então, f (g(1)) + g (f (-1)) é igual a:

a) $\frac{2}{3}$
b) $-\frac{1}{3}$
c) $-\frac{3}{4}$
d) $\frac{3}{2}$
e) $-\frac{4}{3}$

Answer 1

Use a definição (h = função composta)

$\mathsf{~~h_1(x)=(g\circ f)(x)=g(f(x))}\\\\\mathsf{~~h_2(x)=(f\circ g)(x)=f(g(x))}$

Se x₁ = 1 e x₂ = -1, então

$\mathsf{~~h_1=(g\circ f)(x_2)=(g\circ f)(-1)=g(f(-1))=f(-1)-2=3^{-1}-2}\\\\\mathsf{~~h_2=(f\circ g)(x_1)=(f\circ g)(1)=f(g(1))=3^{g(1)}=3^{1-2}=3^{-1}}$

Fazendo h₁ + h₂ temos

$\mathsf{h_1+h_2=3^{-1}-2+3^{-1}}\\\\\mathsf{\hspace{34}=3^{-1}+3^{-1}-2}\\\\\mathsf{\hspace{34}=\frac{1}{3}+\frac{1}{3}-2}\\\\\mathsf{\hspace{34}=\frac{2}{3}-2}\\\\\mathsf{\hspace{34}=\frac{2}{3}-\frac{6}{3}}\\\\\mathsf{\hspace{34}=-\frac{4}{3}~~~~(resposta)}$