If `f:Rto[-1,1]` where `f(x)=sin((pi)/2[x]),` (where [.] denotes the greatest integer fucntion), then

Options

A. `f(x)` is onto
B. `f(x)` is into
C. `f(x)` is periodic
D. `f(x)` is many one

Explanation

To solve the problem, we need to analyze the function \( f: \mathbb{R} \to [-1, 1] \) defined by \( f(x) = \sin\left(\frac{\pi}{2} [x]\right) \), where \( [x] \) denotes the greatest integer function (also known as the floor function).

### Step 1: Understand the Greatest Integer Function
The greatest integer function \( [x] \) gives the largest integer less than or equal to \( x \). For example:
- If \( x = 3.7 \), then \( [x] = 3 \).
- If \( x = -2.3 \), then \( [x] = -3 \).

### Step 2: Determine the Values of \( f(x) \)
The function can take on different values based on the integer part of \( x \). Since \( [x] \) can take any integer value \( n \), we can express \( f(x) \) as:
\[ f(x) = \sin\left(\frac{\pi}{2} n\right) \]
for \( n = [x] \).

### Step 3: Calculate \( f(x) \) for Different Integer Values
We can evaluate \( f(x) \) for various integer values \( n \):
- If \( n = 0 \): \( f(x) = \sin(0) = 0 \)
- If \( n = 1 \): \( f(x) = \sin\left(\frac{\pi}{2}\right) = 1 \)
- If \( n = 2 \): \( f(x) = \sin(\pi) = 0 \)
- If \( n = 3 \): \( f(x) = \sin\left(\frac{3\pi}{2}\right) = -1 \)
- If \( n = 4 \): \( f(x) = \sin(2\pi) = 0 \)

### Step 4: Identify the Range of \( f(x) \)
From the calculations above, we see that the possible values of \( f(x) \) are:
- \( 0 \) (from \( n = 0, 2, 4, \ldots \))
- \( 1 \) (from \( n = 1 \))
- \( -1 \) (from \( n = 3 \))

Thus, the range of \( f(x) \) is \( \{ -1, 0, 1 \} \).

### Step 5: Determine the Type of Function
1. **Onto (Surjective)**: A function is onto if every element in the codomain has a pre-image in the domain. The codomain is \( [-1, 1] \), but the range is \( \{ -1, 0, 1 \} \). Since not all values in the codomain are achieved (e.g., \( 0.5 \) is not in the range), the function is not onto.

2. **Into (Injective)**: A function is into if it does not map distinct elements in the domain to the same element in the codomain. Here, multiple values of \( x \) can yield the same \( f(x) \) (e.g., all \( x \) in the interval \( [0, 1) \) will yield \( f(x) = 0 \)). Thus, the function is not injective.

3. **Periodic**: A function is periodic if there exists a positive number \( p \) such that \( f(x + p) = f(x) \) for all \( x \). The function \( f(x) \) is periodic with period \( 2 \) because \( [x + 2] = [x] + 2 \), leading to the same sine values.

4. **Many-One**: A function is many-one if two or more different inputs can produce the same output. Since multiple values of \( x \) can yield the same \( f(x) \), the function is many-one.

### Conclusion
- The function \( f(x) \) is **not onto**.
- The function \( f(x) \) is **not into**.
- The function \( f(x) \) is **periodic**.
- The function \( f(x) \) is a **many-one function**.