If f(x) = x - `(1)/(x)` , then the value of f(x) + `f((1)/(x))` is :

Options

A. 0
B. 2x
C. `(2)/(x)`
D. None of these

Answer

To solve the problem, we need to find the value of \( f(x) + f\left(\frac{1}{x}\right) \) given that \( f(x) = x - \frac{1}{x} \).

### Step-by-Step Solution:

1. **Find \( f(x) \)**:
\[ f(x) = x - \frac{1}{x} \]

2. **Find \( f\left(\frac{1}{x}\right) \)**:
We need to substitute \( \frac{1}{x} \) into the function:
\[ f\left(\frac{1}{x}\right) = \frac{1}{x} - \frac{1}{\left(\frac{1}{x}\right)} = \frac{1}{x} - x \]
Simplifying this gives:
\[ f\left(\frac{1}{x}\right) = \frac{1}{x} - x \]

3. **Add \( f(x) \) and \( f\left(\frac{1}{x}\right) \)**:
Now, we add the two results:
\[ f(x) + f\left(\frac{1}{x}\right) = \left(x - \frac{1}{x}\right) + \left(\frac{1}{x} - x\right) \]

4. **Combine like terms**:
When we combine the terms:
\[ f(x) + f\left(\frac{1}{x}\right) = x - \frac{1}{x} + \frac{1}{x} - x \]
Notice that \( x \) and \( -x \) cancel each other out, and \( -\frac{1}{x} \) and \( \frac{1}{x} \) also cancel each other out:
\[ f(x) + f\left(\frac{1}{x}\right) = 0 \]

5. **Final Result**:
Therefore, the value of \( f(x) + f\left(\frac{1}{x}\right) \) is:
\[ \boxed{0} \]