Find whether the following function are one-one or many -one & into or onto if `f:DtoR` where `D` is its domain
`f(x)=1/(sinsqrt(|x|))`

Explanation

To determine whether the function \( f(x) = \frac{1}{\sin(\sqrt{|x|})} \) is one-one or many-one, and whether it is into or onto, we will follow these steps:

### Step 1: Determine if the function is one-one or many-one
To check if the function is one-one, we can analyze its behavior for negative and positive values of \( x \).

1. **Evaluate \( f(-x) \)**:
\[ f(-x) = \frac{1}{\sin(\sqrt{|-x|})} = \frac{1}{\sin(\sqrt{|x|})} = f(x) \]
This shows that \( f(x) = f(-x) \). Therefore, the function takes the same value for both \( x \) and \( -x \).

2. **Conclusion**:
Since \( f(x) = f(-x) \), the function is not one-one. It is many-one because multiple inputs (specifically \( x \) and \( -x \)) yield the same output.

### Step 2: Determine if the function is into or onto
Next, we need to analyze the range of the function and compare it with the codomain \( \mathbb{R} \).

1. **Range of \( f(x) \)**:
- The sine function, \( \sin(\sqrt{|x|}) \), oscillates between 0 and 1. Thus, \( \sin(\sqrt{|x|}) \) is always positive and less than or equal to 1.
- Therefore, \( f(x) = \frac{1}{\sin(\sqrt{|x|})} \) will take values from 1 to \( \infty \) as \( \sin(\sqrt{|x|}) \) approaches 1 (giving \( f(x) = 1 \)) and approaches 0 (giving \( f(x) \to \infty \)).

2. **Conclusion**:
- The range of \( f(x) \) is \( [1, \infty) \).
- The codomain is \( \mathbb{R} \), which includes all real numbers.
- Since the range \( [1, \infty) \) does not cover all of \( \mathbb{R} \), the function is not onto.

3. **Final Conclusion**:
- The function is many-one and into.

### Summary of Results
- The function \( f(x) = \frac{1}{\sin(\sqrt{|x|})} \) is **many-one** and **into**.