If f(x) = `x^(3) - (1)/(x^(3))` then f (x) + f`((1)/(x))` is equal to

Options

A. `2x^(3)`
B. `(2)/(x^(3))`
C. `-1`
D. 0

Answer

To solve the problem, we need to find the expression \( f(x) + f\left(\frac{1}{x}\right) \) where \( f(x) = x^3 - \frac{1}{x^3} \).

### Step-by-Step Solution:

1. **Define the function**:
\[ f(x) = x^3 - \frac{1}{x^3} \]

2. **Find \( f\left(\frac{1}{x}\right) \)**:
Substitute \( \frac{1}{x} \) into the function:
\[ f\left(\frac{1}{x}\right) = \left(\frac{1}{x}\right)^3 - \frac{1}{\left(\frac{1}{x}\right)^3} \]
Simplifying this gives:
\[ f\left(\frac{1}{x}\right) = \frac{1}{x^3} - x^3 \]

3. **Combine \( f(x) \) and \( f\left(\frac{1}{x}\right) \)**:
Now, we need to add \( f(x) \) and \( f\left(\frac{1}{x}\right) \):
\[ f(x) + f\left(\frac{1}{x}\right) = \left(x^3 - \frac{1}{x^3}\right) + \left(\frac{1}{x^3} - x^3\right) \]

4. **Simplify the expression**:
When we add these two expressions, we notice that:
\[ f(x) + f\left(\frac{1}{x}\right) = x^3 - \frac{1}{x^3} + \frac{1}{x^3} - x^3 \]
All terms cancel out:
\[ = 0 \]

### Final Answer:
Thus, we conclude that:
\[ f(x) + f\left(\frac{1}{x}\right) = 0 \]