For `x inR - {1}`, the function f(x ) satisfies `f(x) + 2f(1/(1-x))=x`. Find `f(2)`.

Answer

To solve for \( f(2) \) given the equation \( f(x) + 2f\left(\frac{1}{1-x}\right) = x \), we will follow these steps:

### Step 1: Substitute \( x = 2 \)
Start by substituting \( x = 2 \) into the given equation.

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

This simplifies to:

\[ f(2) + 2f(-1) = 2 \quad \text{(Equation 1)} \]

### Step 2: Substitute \( x = -1 \)
Next, substitute \( x = -1 \) into the original equation.

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

This simplifies to:

\[ f(-1) + 2f\left(\frac{1}{2}\right) = -1 \quad \text{(Equation 2)} \]

### Step 3: Substitute \( x = \frac{1}{2} \)
Now, substitute \( x = \frac{1}{2} \) into the original equation.

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

This simplifies to:

\[ f\left(\frac{1}{2}\right) + 2f(2) = \frac{1}{2} \quad \text{(Equation 3)} \]

### Step 4: Solve the equations
Now we have three equations:

1. \( f(2) + 2f(-1) = 2 \) (Equation 1)
2. \( f(-1) + 2f\left(\frac{1}{2}\right) = -1 \) (Equation 2)
3. \( f\left(\frac{1}{2}\right) + 2f(2) = \frac{1}{2} \) (Equation 3)

From Equation 1, we can express \( f(-1) \):

\[ f(-1) = \frac{2 - f(2)}{2} \]

### Step 5: Substitute \( f(-1) \) into Equation 2
Substituting \( f(-1) \) into Equation 2:

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

Multiply through by 2 to eliminate the fraction:

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

This simplifies to:

\[ 4f\left(\frac{1}{2}\right) = f(2) - 4 \quad \text{(Equation 4)} \]

### Step 6: Substitute \( f\left(\frac{1}{2}\right) \) into Equation 3
Now substitute \( f\left(\frac{1}{2}\right) \) from Equation 4 into Equation 3:

From Equation 4, we have:

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

Substituting this into Equation 3:

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

Multiply through by 4 to eliminate the fraction:

\[ f(2) - 4 + 8f(2) = 2 \]

Combine like terms:

\[ 9f(2) - 4 = 2 \]

### Step 7: Solve for \( f(2) \)
Add 4 to both sides:

\[ 9f(2) = 6 \]

Divide by 9:

\[ f(2) = \frac{6}{9} = \frac{2}{3} \]

### Final Answer
Thus, the value of \( f(2) \) is:

\[ \boxed{\frac{2}{3}} \]