Check whether the following functions is/are many-one or one-one & into or onto both solve (i) `f(x)=tan(2sinx)` (ii) `f(x)=tan(sin x)`

Explanation

To determine whether the given functions are one-one or many-one, and whether they are onto or into, we will analyze each function step by step.

### Function (i): \( f(x) = \tan(2\sin x) \)

1. **Check if the function is many-one or one-one:**
- The sine function, \( \sin x \), is periodic with a period of \( 2\pi \). Therefore, \( 2\sin x \) also has a period of \( 2\pi \).
- Since \( \tan \) is also a periodic function, \( f(x) = \tan(2\sin x) \) will repeat its values for \( x + 2\pi \).
- Thus, \( f(x + 2\pi) = \tan(2\sin(x + 2\pi)) = \tan(2\sin x) = f(x) \).
- Therefore, the function is **many-one**.

2. **Check if the function is onto or into:**
- The range of \( 2\sin x \) is from \(-2\) to \(2\) because \( \sin x \) varies from \(-1\) to \(1\).
- The tangent function \( \tan(\theta) \) is defined for \( \theta \) in the interval \( (-\frac{\pi}{2}, \frac{\pi}{2}) \) and takes values from \(-\infty\) to \(+\infty\).
- However, since \( 2\sin x \) only takes values between \(-2\) and \(2\), the output of \( f(x) = \tan(2\sin x) \) will be limited to \( \tan(\theta) \) where \( \theta \in [-2, 2] \).
- Thus, the range of \( f(x) \) is not all real numbers, meaning the function is **into**.

### Function (ii): \( f(x) = \tan(\sin x) \)

1. **Check if the function is many-one or one-one:**
- Similar to the first function, \( \sin x \) is periodic with a period of \( 2\pi \).
- Therefore, \( f(x + 2\pi) = \tan(\sin(x + 2\pi)) = \tan(\sin x) = f(x) \).
- Hence, this function is also **many-one**.

2. **Check if the function is onto or into:**
- The range of \( \sin x \) is from \(-1\) to \(1\).
- The tangent function \( \tan(\theta) \) is defined for \( \theta \) in the interval \( (-\frac{\pi}{2}, \frac{\pi}{2}) \) and takes values from \(-\infty\) to \(+\infty\).
- However, since \( \sin x \) only takes values between \(-1\) and \(1\), the output of \( f(x) = \tan(\sin x) \) will be limited to \( \tan(\theta) \) where \( \theta \in [-1, 1] \).
- Thus, the range of \( f(x) \) is not all real numbers, meaning the function is **into**.

### Summary:
- For \( f(x) = \tan(2\sin x) \):
- Many-one
- Into
- For \( f(x) = \tan(\sin x) \):
- Many-one
- Into