Let `f(x)={{:(x^(2) sin ((pix)/(2)),-1 lt x lt 1, x ne 0),(x|x|, x gt 1 or x le -1):}` . Then ,

Options

A. f(x) is an odd function
B. f(x) is an even function
C. f(x) is neither odd nor even
D. none of these

Answer

To determine whether the function \( f(x) \) is even, odd, or neither, we need to analyze the given piecewise function:

\[ f(x) = \begin{cases} x^2 \sin\left(\frac{\pi x}{2}\right) & \text{for } -1 < x < 1, \, x \neq 0 \\ x |x| & \text{for } x \geq 1 \text{ or } x \leq -1 \end{cases} \]

### Step 1: Define the conditions for even and odd functions
- A function \( f(x) \) is **even** if \( f(-x) = f(x) \) for all \( x \).
- A function \( f(x) \) is **odd** if \( f(-x) = -f(x) \) for all \( x \).

### Step 2: Find \( f(-x) \)
We will find \( f(-x) \) for both cases of the piecewise function.

1. **For \( -1 < x < 1 \)**:
- Here, \( -x \) will also lie in the interval \( -1 < -x < 1 \).
- Thus,
\[ f(-x) = (-x)^2 \sin\left(\frac{\pi (-x)}{2}\right) = x^2 \sin\left(-\frac{\pi x}{2}\right) = -x^2 \sin\left(\frac{\pi x}{2}\right) \]

2. **For \( x \geq 1 \) or \( x \leq -1 \)**:
- If \( x \geq 1 \), then \( -x \leq -1 \).
- Thus,
\[ f(-x) = -x |x| = -x^2 \quad \text{(since } |x| = x \text{ for } x \geq 1\text{)} \]

- If \( x \leq -1 \), then \( -x \geq 1 \).
- Thus,
\[ f(-x) = -x |x| = -x^2 \quad \text{(since } |x| = -x \text{ for } x \leq -1\text{)} \]

### Step 3: Compare \( f(-x) \) and \( -f(x) \)
Now we will compare \( f(-x) \) with \( -f(x) \).

1. **For \( -1 < x < 1 \)**:
- We have \( f(x) = x^2 \sin\left(\frac{\pi x}{2}\right) \).
- Therefore,
\[ -f(x) = -x^2 \sin\left(\frac{\pi x}{2}\right) \]
- Since \( f(-x) = -x^2 \sin\left(\frac{\pi x}{2}\right) \), we see that \( f(-x) = -f(x) \).

2. **For \( x \geq 1 \) or \( x \leq -1 \)**:
- We have \( f(x) = x |x| = x^2 \).
- Therefore,
\[ -f(x) = -x^2 \]
- Since \( f(-x) = -x^2 \), we see that \( f(-x) = -f(x) \).

### Conclusion
Since \( f(-x) = -f(x) \) holds true for all cases, we conclude that the function \( f(x) \) is an **odd function**.

### Final Answer
The function \( f(x) \) is an odd function.