Question

Dadas as funções f(x) = 3x+4 e g(x) = 2x-1. Calcule x de modo que g-¹ (f(x)) = f (g(2))

Answer 1

$f(x)=3x+4\\g(x)=2x-1$

Calculando a inversa de g:

$g(x)=2x-1\\x=2g^{-1}(x)-1\\x+1=2g^{-1}(x)\\(x+1)/2=g^{-1}(x)$
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$g^{-1}(x)=(x+1)/2\\g^{-1}(f(x))=(f(x)+1)/2\\g^{-1}(f(x))=(3x+4+1)/2\\g^{-1}(f(x))=(3x+5)/2$
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Calculando f(g(2)):

$g(x)=2x-1\\g(2)=2.2-1\\g(2)=4-1\\g(2)=3\\\\f(g(2))=f(3)\\f(g(2))=3.3+4\\f(g(2))=9+4\\f(g(2))=13$
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Finalmente:

$g^{-1}(f(x))=f(g(2))\\g^{-1}(f(x))=13\\(3x+5)/2=13\\3x+5=2.13\\3x+5=26\\3x=26-5\\3x=21\\x=21/3\\x=7$