Matemática, perguntado por builtfences, 11 meses atrás

Se g(x) = 1-x e (fog)(x) = \frac{1-x}{x}, com x≠0, então f(\frac{4}{3}) vale:

a) 1

b) \frac{1}{4}

c) 4

d) -\frac{1}{4}

e) -4

Soluções para a tarefa

Respondido por luizfellipe95
1

Resposta:

fog(x)=

x

1−x

f(g(x))=

x

1−x

f(1−x)=

x

1−x

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

\begin{lgathered}1-x=k\\x=1-k\end{lgathered}

1−x=k

x=1−k

Logo:

\begin{lgathered}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}\end{lgathered}

f(1−x)=

x

1−x

f(k)=

1−k

1−(1−k)

f(k)=

1−k

1−1+k

f(k)=

1−k

k

Trocando k por x:

\boxed{\boxed{f(x)=\dfrac{x}{1-x}}}

f(x)=

1−x

x

________________________

\begin{lgathered}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}}\end{lgathered}

f(x)=

1−x

x

f(

3

4

)=

1−(

3

4

)

(

3

4

)

f(

3

4

)=

(−

3

1

)

(

3

4

)

f(

3

4

)=−

(

3

1

)

(

3

4

)

f(

3

4

)=−

3

4

1

3

f(

3

4

)=−4

______________________________________

Se não quiser achar f(x) primeiro, podemos achar o valor de x que faça com que 1 - x seja igual a 4/3

\begin{lgathered}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}}}\end{lgathered}

f(1−x)=f(

3

4

)

1−x=

3

4

x=1−

3

4

x=−

3

1

Então:

\begin{lgathered}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}}\end{lgathered}

f(1−x)=

x

1−x

f(1−[−

3

1

])=

(−

3

1

)

1−(−

3

1

)

f(

3

4

)=−4


builtfences: Eu não sei se o problema está na sua formatação ou no meu computador, porém não consigo entender sua resposta :(
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