`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 \( f(x) = x \cdot \text{sgn}(x^2) \).

### Step 1: Understand the Signum Function
The signum function, denoted as \(\text{sgn}(x)\), is defined as:
- \(\text{sgn}(x) = 1\) if \(x > 0\)
- \(\text{sgn}(x) = -1\) if \(x < 0\)
- \(\text{sgn}(x) = 0\) if \(x = 0\)

### Step 2: Determine the Value of \( \text{sgn}(x^2) \)
Since \(x^2\) is always non-negative:
- If \(x \neq 0\), \(x^2 > 0\) and thus \(\text{sgn}(x^2) = 1\).
- If \(x = 0\), then \(\text{sgn}(x^2) = 0\).

### Step 3: Evaluate \( f(x) \) for Different Cases
Now we can evaluate \( f(x) \) based on the value of \( x \):

1. **Case 1: \( x > 0 \)**
- Here, \( \text{sgn}(x^2) = 1 \).
- Therefore, \( f(x) = x \cdot 1 = x \).

2. **Case 2: \( x < 0 \)**
- Here, \( \text{sgn}(x^2) = 1 \).
- Therefore, \( f(x) = x \cdot 1 = x \).

3. **Case 3: \( x = 0 \)**
- Here, \( \text{sgn}(x^2) = 0 \).
- Therefore, \( f(x) = 0 \cdot 0 = 0 \).

### Step 4: Summarize the Results
From the evaluations:
- If \( x > 0 \), then \( f(x) = x \).
- If \( x < 0 \), then \( f(x) = x \).
- If \( x = 0 \), then \( f(x) = 0 \).

### Conclusion
Thus, we can conclude:
- \( f(x) = x \) for \( x > 0 \)
- \( f(x) = x \) for \( x < 0 \)
- \( f(x) = 0 \) for \( x = 0 \)

### Final Answer
All options are correct:
1. \( f(x) = x \) if \( x > 0 \)
2. \( f(x) = x \) if \( x < 0 \)
3. \( f(x) = 0 \) if \( x = 0 \)
4. None of these (not applicable)