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Aerioeus · Answer 1 · 2023-08-29T21:08:54+0000

Answer:

(f o g)(x) = x

Step-by-step explanation:

Given f(x) and g(x) defined below:

$\begin{gathered} f(x)=(1-x)/(x) \\ g(x)=(1)/(1+x) \end{gathered}$

The composition (f o g)(x) is obtained below:

$\begin{gathered} (f\circ g)(x)=f\lbrack g(x)\rbrack \\ f(x)=(1-x)/(x)\implies f\lbrack g(x)\rbrack=(1-g(x))/(g(x)) \end{gathered}$

Substitute g(x) into the expression and simplify:

$\begin{gathered} f\lbrack g(x)\rbrack=(1-g(x))/(g(x))=\lbrack1-g(x)\rbrack/ g(x) \\ =(1-(1)/(1+x))/((1)/(1+x)) \\ \text{ Take the LCM in the first bracket} \\ =(1(1+x)-1)/(1+x)/(1)/(1+x)\text{ } \\ \text{Open the bracket} \\ =(1+x-1)/(1+x)/(1)/(1+x)\text{ } \\ =(x)/(1+x)*(1+x)/(1)\text{ } \\ =x \end{gathered}$

Therefore, the composition (f o g)(x) is x.

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