`f(x)=x sgn (x^(2))` should be

Options

A. `x, x gt 0`
B. `x, x lt 0`
C. `0,x=0`
D. none of these

Answer

To solve the problem, we need to analyze the function \( f(x) = x \cdot \text{sgn}(x^2) \) and determine its behavior based on the value of \( x \).

### Step-by-Step Solution:

1. **Understanding the Signum Function**:
The signum function, denoted as \(\text{sgn}(x)\), is defined as follows:
- \(\text{sgn}(x) = 1\) if \( x > 0 \)
- \(\text{sgn}(x) = 0\) if \( x = 0 \)
- \(\text{sgn}(x) = -1\) if \( x < 0 \)

2. **Evaluating \( \text{sgn}(x^2) \)**:
Since \( x^2 \) is always non-negative (i.e., \( x^2 \geq 0 \)), we can analyze \( \text{sgn}(x^2) \):
- If \( x > 0 \), then \( x^2 > 0 \) and thus \(\text{sgn}(x^2) = 1\).
- If \( x = 0 \), then \( x^2 = 0 \) and thus \(\text{sgn}(x^2) = 0\).
- If \( x < 0 \), then \( x^2 > 0 \) and thus \(\text{sgn}(x^2) = 1\).

3. **Substituting into the Function**:
Now we substitute the values of \(\text{sgn}(x^2)\) into the function \( f(x) \):
- For \( x > 0 \):
\[ f(x) = x \cdot \text{sgn}(x^2) = x \cdot 1 = x \]
- For \( x = 0 \):
\[ f(x) = x \cdot \text{sgn}(x^2) = 0 \cdot 0 = 0 \]
- For \( x < 0 \):
\[ f(x) = x \cdot \text{sgn}(x^2) = x \cdot 1 = x \]

4. **Conclusion**:
From the evaluations, we can summarize the function \( f(x) \):
- \( f(x) = x \) when \( x > 0 \)
- \( f(x) = 0 \) when \( x = 0 \)
- \( f(x) = x \) when \( x < 0 \)

Thus, the function can be expressed as:
\[ f(x) = \begin{cases} x & \text{if } x > 0 \\ 0 & \text{if } x = 0 \\ x & \text{if } x < 0 \end{cases} \]

### Final Answer:
The correct options are:
- \( f(x) = x \) when \( x > 0 \)
- \( f(x) = 0 \) when \( x = 0 \)
- \( f(x) = x \) when \( x < 0 \)