Find whether the following function are one-one or many -one & into or onto if `f:DtoR` where `D` is its domain
`f(x)=sin4x:(-(pi)/4,(pi)/4)to(-1,1)`

Explanation

To determine whether the function \( f(x) = \sin(4x) \) defined from the domain \( D = \left(-\frac{\pi}{4}, \frac{\pi}{4}\right) \) to the codomain \( R = (-1, 1) \) is one-one or many-one, and whether it is into or onto, we can follow these steps:

### Step 1: Analyze the Function
The function given is \( f(x) = \sin(4x) \). The first step is to understand how this function behaves within the specified domain.

### Step 2: Determine the Range of the Function
1. **Calculate the endpoints of the domain:**
- At \( x = -\frac{\pi}{4} \):
\[ f\left(-\frac{\pi}{4}\right) = \sin\left(4 \times -\frac{\pi}{4}\right) = \sin(-\pi) = 0 \]
- At \( x = \frac{\pi}{4} \):
\[ f\left(\frac{\pi}{4}\right) = \sin\left(4 \times \frac{\pi}{4}\right) = \sin(\pi) = 0 \]

2. **Evaluate the function within the domain:**
- The function \( f(x) = \sin(4x) \) is periodic and oscillates between -1 and 1. Since we are looking at the interval \( \left(-\frac{\pi}{4}, \frac{\pi}{4}\right) \), we can find the maximum and minimum values of \( f(x) \) in this interval.
- The maximum value occurs at \( x = \frac{1}{4} \) (where \( 4x = \frac{\pi}{2} \)):
\[ f\left(\frac{1}{4}\right) = \sin\left(\frac{\pi}{2}\right) = 1 \]
- The minimum value occurs at \( x = -\frac{1}{4} \) (where \( 4x = -\frac{\pi}{2} \)):
\[ f\left(-\frac{1}{4}\right) = \sin\left(-\frac{\pi}{2}\right) = -1 \]

Thus, the range of \( f(x) \) over the interval \( \left(-\frac{\pi}{4}, \frac{\pi}{4}\right) \) is \( (-1, 1) \).

### Step 3: Determine if the Function is One-One or Many-One
To check if the function is one-one or many-one, we can use the horizontal line test:
- The function \( \sin(4x) \) oscillates and will take the same value for different inputs within the interval \( \left(-\frac{\pi}{4}, \frac{\pi}{4}\right) \).
- For example, \( f(0) = 0 \) and \( f(-\frac{\pi}{8}) = \sin(-\frac{\pi}{2}) = -1 \) and \( f(\frac{\pi}{8}) = \sin(\frac{\pi}{2}) = 1 \).
- Since the function takes the same value for different inputs, it is many-one.

### Step 4: Determine if the Function is Into or Onto
- The codomain is \( (-1, 1) \) and the range is also \( (-1, 1) \).
- Since the range matches the codomain, the function is onto.

### Conclusion
The function \( f(x) = \sin(4x) \) is many-one and onto.