Find whether the following function are one-one or many -one & into or onto if `f:DtoR` where `D` is its domain
`f(x)=x cos x`

Explanation

To determine whether the function \( f(x) = x \cos x \) is one-one or many-one, and whether it is into or onto, we will follow a systematic approach.

### Step 1: Determine if the function is one-one or many-one

To check if the function is one-one, we need to see if different inputs can produce the same output.

1. **Evaluate \( f\left(\frac{\pi}{2}\right) \)**:
\[ f\left(\frac{\pi}{2}\right) = \frac{\pi}{2} \cdot \cos\left(\frac{\pi}{2}\right) = \frac{\pi}{2} \cdot 0 = 0 \]

2. **Evaluate \( f\left(\frac{5\pi}{2}\right) \)**:
\[ f\left(\frac{5\pi}{2}\right) = \frac{5\pi}{2} \cdot \cos\left(\frac{5\pi}{2}\right) \]
Since \( \cos\left(\frac{5\pi}{2}\right) = \cos\left(2\pi + \frac{\pi}{2}\right) = \cos\left(\frac{\pi}{2}\right) = 0 \),
\[ f\left(\frac{5\pi}{2}\right) = \frac{5\pi}{2} \cdot 0 = 0 \]

Since \( f\left(\frac{\pi}{2}\right) = f\left(\frac{5\pi}{2}\right) = 0 \), we have found two different inputs that yield the same output. Thus, the function is **many-one**.

### Step 2: Determine if the function is into or onto

Next, we need to find the range of the function to determine if it is into or onto. The function's codomain is \( \mathbb{R} \).

1. **Analyze the behavior of \( f(x) = x \cos x \)**:
- As \( x \) approaches \( \infty \), \( \cos x \) oscillates between -1 and 1. Therefore, \( f(x) \) oscillates between \( -x \) and \( x \).
- As \( x \) approaches \( -\infty \), \( f(x) \) also oscillates between \( -x \) and \( x \).

2. **Determine the range**:
- Since \( \cos x \) oscillates between -1 and 1, \( f(x) \) can take values from \( -\infty \) to \( \infty \) as \( x \) varies. Thus, the range of \( f(x) \) is \( (-\infty, \infty) \).

Since the range of \( f(x) \) is \( \mathbb{R} \) and the codomain is also \( \mathbb{R} \), the function is **onto**.

### Conclusion

- The function \( f(x) = x \cos x \) is **many-one**.
- The function is **onto**.