Function `f[(1)/(2)pi, (3)/(2)pi] rarr [-1, 1], f(x) = cos x` is

Options

A. Many-one onto
B. Onto
C. One-one onto
D. Many-one into

Answer

To determine the nature of the function \( f(x) = \cos x \) with the domain \( \left[\frac{\pi}{2}, \frac{3\pi}{2}\right] \) and the co-domain \( [-1, 1] \), we will analyze whether it is one-one, onto, many-one, or into.

### Step-by-Step Solution:

1. **Identify the Function and Its Domain:**
- We have the function \( f(x) = \cos x \).
- The domain is \( \left[\frac{\pi}{2}, \frac{3\pi}{2}\right] \).

2. **Graph the Function:**
- The cosine function oscillates between -1 and 1.
- For the specified domain, the graph of \( \cos x \) will start at \( \cos\left(\frac{\pi}{2}\right) = 0 \) and end at \( \cos\left(\frac{3\pi}{2}\right) = 0 \).
- The minimum value occurs at \( x = \pi \), where \( \cos(\pi) = -1 \).

3. **Determine the Range of the Function:**
- From the graph, we can see that as \( x \) moves from \( \frac{\pi}{2} \) to \( \frac{3\pi}{2} \), \( f(x) \) decreases from 0 to -1 and then returns to 0.
- Therefore, the range of \( f(x) \) over the given domain is \( [-1, 0] \).

4. **Check if the Function is One-One:**
- A function is one-one if no two different inputs produce the same output.
- In this case, since the graph of \( f(x) \) is decreasing from \( 0 \) to \( -1 \) and then increasing back to \( 0 \), it intersects the horizontal line \( y = c \) (for \( c \) in the range) at two points.
- Therefore, the function is **many-one**.

5. **Check if the Function is Onto:**
- A function is onto if every element in the co-domain is mapped by at least one element in the domain.
- The co-domain is \( [-1, 1] \) while the range is \( [-1, 0] \).
- Since the range does not cover the entire co-domain (it does not reach values greater than 0), the function is **not onto**.

6. **Determine if the Function is Into:**
- A function is into if it does not cover the entire co-domain.
- Since the range \( [-1, 0] \) does not cover the co-domain \( [-1, 1] \), the function is **into**.

### Conclusion:
Based on the analysis, we conclude that the function \( f(x) = \cos x \) defined on the domain \( \left[\frac{\pi}{2}, \frac{3\pi}{2}\right] \) and co-domain \( [-1, 1] \) is **many-one and into**.

### Final Answer:
The function is many-one into.