Question

Sejam as funções f(x) = x²-2x+1 e g(x) = 2x+1, calcule:
a) (fog)(x)

b) (gof)(2)

Answer 1

Lembre:

$fog(x)=f(g(x))\\gof(x)=g(f(x))$
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a) fog(x)

$fog(x)=f(g(x))$

$f(x)=x^{2}-2x+1\\f(g(x))=[g(x)]^{2}-2[g(x)]+1\\fog(x)=[2x+1]^{2}-2[2x+1]+1\\fog(x)=[(2x)^{2}+2*2x*1+1^{2}]-4x-2+1\\fog(x)=[4x^{2}+4x+1]-4x-1\\fog(x)=4x^{2}+4x-4x+1-1\\fog(x)=4x^{2}$

b)

$gof(2)=g(f(2))$

$f(x)=x^{2}-2x+1\\f(2)=2^{2}-2*2+1\\f(2)=4-4+1\\f(2)=1$

$g(f(2))=g(1)\\g(x)=2x+1\\g(1)=2*1+1\\g(1)=2+1\\g(1)=3$

$\boxed{\boxed{gof(2)=3}}$