Question

For $x\in R-\{0,1\}$ , let $f_1(x)=\frac{1}{x},f_2(x)=1-x$ and $f_3(x)=\frac{1}{1-x}$ be three given functions. If a function, J(x) satisfies $(f_2{}^{\circ }J{{}^{\circ }f}_1)(x)=f_3(x)$ then J(x) is equal to :-

• A. $f_3(x)$
• B. $f_1(x)$
• C. $f_2(x)$
• D. $\frac{1}{x}{f}_3(x)$

Answer 1

Answer: (A)

Given $f_1\left(x\right)=\frac{1}{x},f_2\left(x\right)=1-x$ and $f_3\left(x\right)=\frac{1}{1-x}$
$\left(f_2\circ J\circ f_1\right)\left(x\right)=f_3\left(x\right)$
$f_2\circ \left(J\left(f_1\left(x\right)\right)\right)=f_3\left(x\right)$
$f_2\circ \left(J\left(\frac{1}{x}\right)\right)=\frac{1}{1-x}$
$1-J\left(\frac{1}{x}\right)=\frac{1}{1-x}$
$J\left(\frac{1}{x}\right)=1-\frac{1}{1-x}=\frac{-x}{1-x}=\frac{x}{x-1}$
Now $x\to \frac{1}{x}$
$J\left(x\right)=\frac{\frac{1}{x}}{\frac{1}{x}-1}=\frac{1}{1-x}=f_3\left(x\right)$