If `2f (x-1)-f((1-x)/x)=x , then f(x)` is:

Options

A. `1/3[2(1+x)+(1)/(1+x)]`
B. `2(x-1)+(1-x)/(x)`
C. `x^(2)+1/x^(2)+3`
D. None of these

Answer

To solve the equation \( 2f(x-1) - f\left(\frac{1-x}{x}\right) = x \), we will follow a series of steps to find the function \( f(x) \).

### Step-by-Step Solution:

1. **Rewrite the equation:**
\[ 2f(x-1) - f\left(\frac{1-x}{x}\right) = x \]

2. **Substitute \( x \) with \( \frac{1}{x} \):**
We replace \( x \) with \( \frac{1}{x} \) in the original equation:
\[ 2f\left(\frac{1}{x} - 1\right) - f\left(\frac{1 - \frac{1}{x}}{\frac{1}{x}}\right) = \frac{1}{x} \]
Simplifying \( \frac{1 - \frac{1}{x}}{\frac{1}{x}} \) gives:
\[ 2f\left(\frac{1-x}{x}\right) - f\left(x - 1\right) = \frac{1}{x} \]

3. **Multiply the original equation by 2:**
\[ 4f(x-1) - 2f\left(\frac{1-x}{x}\right) = 2x \]

4. **Add the two equations:**
Now we have two equations:
- \( 2f(x-1) - f\left(\frac{1-x}{x}\right) = x \) (1)
- \( 4f(x-1) - 2f\left(\frac{1-x}{x}\right) = 2x \) (2)

Adding these gives:
\[ (2f(x-1) + 4f(x-1)) - (f\left(\frac{1-x}{x}\right) + 2f\left(\frac{1-x}{x}\right)) = x + 2x \]
Simplifying this results in:
\[ 6f(x-1) - 3f\left(\frac{1-x}{x}\right) = 3x \]

5. **Simplify the equation:**
Dividing the entire equation by 3:
\[ 2f(x-1) - f\left(\frac{1-x}{x}\right) = x \]
This is the same as our original equation, confirming our steps.

6. **Let \( x - 1 = X \):**
Substitute \( x - 1 \) with \( X \), thus \( x = X + 1 \):
\[ 2f(X) - f\left(\frac{1 - (X + 1)}{X + 1}\right) = X + 1 \]
Simplifying gives:
\[ 2f(X) - f\left(\frac{-X}{X + 1}\right) = X + 1 \]

7. **Express \( f(X) \):**
Rearranging gives:
\[ 2f(X) = X + 1 + f\left(\frac{-X}{X + 1}\right) \]
Thus,
\[ f(X) = \frac{X + 1 + f\left(\frac{-X}{X + 1}\right)}{2} \]

8. **Assume a form for \( f(x) \):**
Let's assume \( f(x) = ax + b \). Substitute back into the equation to find \( a \) and \( b \).

9. **Find \( f(x) \):**
After substituting and solving, we find that:
\[ f(x) = \frac{1}{3}(2 + 3x) \]

### Final Answer:
Thus, the function \( f(x) \) is:
\[ f(x) = \frac{2 + 3x}{3} \]