If `f(x)=x^(3)`, which of the following must be true ?

Options

A. `f(-x)=f(x)`
B. `f(-x)=-f(-x)`
C. `f(-x)=-f(x)`
D. `f(x)=-f(x)`

Explanation

To solve the problem, we need to evaluate the function \( f(x) = x^3 \) at \( -x \) and determine which of the provided statements is true.

### Step-by-Step Solution:

1. **Identify the function**: We have the function defined as \( f(x) = x^3 \).

2. **Substitute \( -x \) into the function**: We need to find \( f(-x) \).
\[ f(-x) = (-x)^3 \]

3. **Calculate \( (-x)^3 \)**:
\[ (-x)^3 = -x^3 \]
This is because raising a negative number to an odd power results in a negative number.

4. **Relate \( f(-x) \) to \( f(x) \)**: Since we know \( f(x) = x^3 \), we can express \( -x^3 \) in terms of \( f(x) \):
\[ f(-x) = -x^3 = -f(x) \]

5. **Conclusion**: We have found that \( f(-x) = -f(x) \). This means that the function \( f(x) = x^3 \) satisfies the property of being an odd function.

6. **Select the correct option**: Among the options provided, the statement that must be true is:
\[ \text{Option 3: } f(-x) = -f(x) \]