Question

For$x\in R-1$, the function $f\left(x\right)$satisfies$f\left(x\right)+2f\left[\frac{1}{(1-x)}\right]=x$. then what is the value of $f\left(2\right)$?

• A. $\frac{2}{3}$
• B. $\frac{1}{3}$
• C. $\frac{4}{3}$
• D. $\frac{7}{3}$

Answer 1

The correct option is A

$\frac{2}{3}$

The explanation for correct option:

Step1. Construction :

Given that the function $f\left(x\right)$satisfies$f\left(x\right)+2f\left[\frac{1}{\left(1-x\right)}\right]=x$.

Put$x=2$, we get $f\left(2\right)+2f(-1)=2...........\left(1\right)$

Put$x=-1$, we get $f(-1)+2f\left(\frac{1}{2}\right)=-1.........\left(2\right)$

Put$x=1/2$, we get $f\left(\frac{1}{2}\right)+2f\left(2\right)=\frac{1}{2}..........\left(3\right)$

From equation $\left(1\right),$ $f\left(2\right)=2–2f(-1)$

Now, put the value of $f(-1)$from the equation$\left(2\right)$. We get

$f\left(2\right)=2–2\left(-1–2f\left(\frac{1}{2}\right)\right)$

$=2+2+4f\left(\frac{1}{2}\right)$

Step2. Find the value of $f\left(2\right)$ :

Now, put the value of $f\left(\frac{1}{2}\right)$from the equation$\left(3\right)$. We get

$f\left(2\right)=4+4\left[\left(\frac{1}{2}\right)-2f\left(2\right)\right]$

To simply this we get $f\left(2\right)=4+2–8f\left(2\right)$

$$\begin{gathered} \Rightarrow 9f\left(2\right)=6 \\ \Rightarrow f\left(2\right)=\frac{6}{9} \\ \Rightarrow f\left(2\right)=\frac{2}{3} \end{gathered}$$

Hence, the correct option is (A)