Function `f : [(pi)/(2), (3pi)/(2)] rarr [-1, 1], f(x) = sin 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 \( f: \left[\frac{\pi}{2}, \frac{3\pi}{2}\right] \to [-1, 1] \) defined by \( f(x) = \sin x \), we will check if it is a one-to-one function (1-1) and whether it is onto (surjective).

### Step 1: Check if the function is one-to-one (1-1)

A function is one-to-one if it never assigns the same value to two different domain elements.

1. **Graph the function**: We need to graph \( f(x) = \sin x \) over the interval \( \left[\frac{\pi}{2}, \frac{3\pi}{2}\right] \).
- At \( x = \frac{\pi}{2} \), \( f\left(\frac{\pi}{2}\right) = 1 \).
- At \( x = \frac{3\pi}{2} \), \( f\left(\frac{3\pi}{2}\right) = -1 \).
- The sine function decreases from 1 to -1 in this interval.

2. **Horizontal Line Test**: To check if the function is one-to-one, we apply the horizontal line test. If any horizontal line intersects the graph of the function more than once, then the function is not one-to-one.
- Since the sine function is decreasing in the interval \( \left[\frac{\pi}{2}, \frac{3\pi}{2}\right] \), any horizontal line will intersect the graph at most once.

Thus, the function \( f(x) = \sin x \) is **one-to-one**.

### Step 2: Check if the function is onto (surjective)

A function is onto if every element in the codomain has a pre-image in the domain.

1. **Determine the range of the function**: The range of \( f(x) = \sin x \) over the interval \( \left[\frac{\pi}{2}, \frac{3\pi}{2}\right] \) is from -1 to 1.
- The minimum value is \( -1 \) (at \( x = \frac{3\pi}{2} \)) and the maximum value is \( 1 \) (at \( x = \frac{\pi}{2} \)).

2. **Compare range with codomain**: The codomain of the function is \( [-1, 1] \).
- Since the range of \( f(x) \) is exactly \( [-1, 1] \), every value in the codomain is achieved by some value in the domain.

Thus, the function \( f(x) = \sin x \) is **onto**.

### Conclusion

Since the function \( f(x) = \sin x \) is both one-to-one and onto, we conclude that it is a **bijective function**.