Question

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Considere a função f(x) = 3x + 4. Calcule o valor de f of of-1(4).

Answer 1

$f\circ f\circ f(x)~=~3\,.\,\left(~f\,(f(x)\,)~\right)+4\\\\\\f\circ f\circ f(x)~=~3\,.\,\left(\,3\,.\,f(x)+4\,\right)+4\\\\\\f\circ f\circ f(x)~=~3\,.\,\left(\,3\,.\,(3x+4)+4\,\right)+4\\\\\\f\circ f\circ f(x)~=~3\,.\,\left(9x+12+4\,\right)+4\\\\\\f\circ f\circ f(x)~=~3\,.\,\left(9x+16\,\right)+4\\\\\\f\circ f\circ f(x)~=~27x+48+4\\\\\\\boxed{f\circ f\circ f(x)~=~27x+52}$

Vamos agora determinar a inversa de fofof(x):

$f\circ f\circ f(x)~=~y~=~27x+52\\\\\\Trocando~"x"~por~"y":\\\\\\x~=~27y+52\\\\\\Isolando~"y":\\\\\\x-52~=~27y\\\\\\\boxed{y~=~\frac{x-52}{27}}$

Calculando fofof^{-1}(4):

$f\circ f\circ f^{-1}(4)~=~\frac{4-52}{27}\\\\\\f\circ f\circ f^{-1}(4)~=~\frac{-48}{27}\\\\\\\boxed{f\circ f\circ f^{-1}(4)~=~-\frac{16}{9}}$