Question

(FURG) Se g(x) = 1-x e fog(x) = 1-x/x  (x diferente de 0), então f (4/3) vale?

a)1
b)1/4
c)4
d)-1/4
e)-4

(POR FAVOR, preciso que me expliquem passo a passo como resolver)
Eu sei que fog (x) = f(g(x)) ---> f(g(x)) = f(1-x), vendo isso cheguei cheguei na seguinte parte:

f(1-x) = 1-x/x  

E depois disso eu não faço a mínima ideia do que fazer.

Answer 1

$fog(x)=\dfrac{1-x}{x}\\\\\\f(g(x))=\dfrac{1-x}{x}\\\\\\f(1-x)=\dfrac{1-x}{x}$

Vamos chamar 1 - x de k, e isolar x:

$1-x=k\\x=1-k$

Logo:

$f(1-x)=\dfrac{1-x}{x}\\\\\\f(k)=\dfrac{1-(1-k)}{1-k}\\\\\\f(k)=\dfrac{1-1+k}{1-k}\\\\\\f(k)=\dfrac{k}{1-k}$

Trocando k por x:

$\boxed{\boxed{f(x)=\dfrac{x}{1-x}}}$
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$f(x)=\dfrac{x}{1-x}\\\\\\f\left(\dfrac{4}{3}\right)=\dfrac{\left(\frac{4}{3}\right)}{1-\left(\frac{4}{3}\right)}\\\\\\f\left(\dfrac{4}{3}\right)=\dfrac{\left(\frac{4}{3}\right)}{\left(-\frac{1}{3}\right)}\\\\\\f\left(\dfrac{4}{3}\right)=-\dfrac{\left(\frac{4}{3}\right)}{\left(\frac{1}{3}\right)}\\\\\\f\left(\dfrac{4}{3}\right)=-\dfrac{4}{3}*\dfrac{3}{1}\\\\\\\boxed{\boxed{f\left(\dfrac{4}{3}\right)=-4}}$
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Se não quiser achar f(x) primeiro, podemos achar o valor de x que faça com que 1 - x seja igual a 4/3

$f(1-x)=f\left(\dfrac{4}{3}\right)\\\\\\1-x=\dfrac{4}{3}\\\\\\x=1-\dfrac{4}{3}\\\\\\\boxed{\boxed{x=-\dfrac{1}{3}}}$

Então:

$f(1-x)=\dfrac{1-x}{x}\\\\\\f\left(1-\left[-\dfrac{1}{3}\right]\right)=\dfrac{1-\left(-\frac{1}{3}\right)}{\left(-\frac{1}{3}\right)}~~~~\therefore~~~~\boxed{\boxed{f\left(\dfrac{4}{3}\right)=-4}}$