Question

Função composta
1-se f(x)=x^2+2x e g(x)=1-3x, determine:
A) fog(x)
B) gof(x)
C) fof(x)
D) gog(x)

Answer 1

Temos as seguintes funções:

$\mathsf{f:~\mathbb{R}\to\mathbb{R}}\\\mathsf{\hspace{18}x\mapsto x^2+2x}\\\\\\\mathsf{g:~\mathbb{R}\to\mathbb{R}}\\\mathsf{\hspace{19}x\mapsto 1-3x}$

Pede-se para definir as seguintes compostas: 

$\mathsf{(f\circ g)(x)=f(g(x))=(g(x))^2+2\cdot(g(x))=(1-3x)^2+2\cdot(1-3x)}\\\\\mathsf{=1-6x+9x^2+2-6x=9x^2-6x-6x+1+2=\underline{9x^2-12x+3}}$


$\mathsf{(g\circ f)(x)=g(f(x))=1-3\cdot(f(x))=1-3\cdot(x^2+2x)=1-3x^2+6x}\\\\\mathsf{=\underline{-3x^2-6x+1}}$ 


$\mathsf{(f\circ f)(x)=f(f(x))=(f(x))^2+2\cdot(f(x))=(x^2+2x)^2+2\cdot (x^2+2x)}\\\\\mathsf{=x^4+2\cdot x^2\cdot 2x+4x^2+2x^2+4x=\underline{x^4+4x^3+6x^2+4x}}$


$\mathsf{(g\circ g)(x)=g(g(x))=1-3\cdot (g(x))=1-3\cdot(1-3x)=1-3+9x}\\\\\mathsf{=\underline{9x-2}}$