Answer:
(f o g)(x) = x
Explanation:
Given f(x) and g(x) defined below:
[tex]\begin{gathered} f(x)=\frac{1-x}{x} \\ g(x)=\frac{1}{1+x} \end{gathered}[/tex]
The composition (f o g)(x) is obtained below:
[tex]\begin{gathered} (f\circ g)(x)=f\lbrack g(x)\rbrack \\ f(x)=\frac{1-x}{x}\implies f\lbrack g(x)\rbrack=\frac{1-g(x)}{g(x)} \end{gathered}[/tex]
Substitute g(x) into the expression and simplify:
[tex]\begin{gathered} f\lbrack g(x)\rbrack=\frac{1-g(x)}{g(x)}=\lbrack1-g(x)\rbrack\div g(x) \\ =(1-\frac{1}{1+x})\div(\frac{1}{1+x}) \\ \text{ Take the LCM in the first bracket} \\ =\frac{1(1+x)-1}{1+x}\div\frac{1}{1+x}\text{ } \\ \text{Open the bracket} \\ =\frac{1+x-1}{1+x}\div\frac{1}{1+x}\text{ } \\ =\frac{x}{1+x}\times\frac{1+x}{1}\text{ } \\ =x \end{gathered}[/tex]
Therefore, the composition (f o g)(x) is x.
