Following each of the following function find whether it is one-one or many-one and also into or onto
`f(x)=2tanx,(pi//2,3pi//2)toR`

Explanation

To determine whether the function \( f(x) = 2\tan x \) defined on the interval \( \left(\frac{\pi}{2}, \frac{3\pi}{2}\right) \) is one-one or many-one, and whether it is into or onto, we will follow these steps:

### Step 1: Understand the Function and Its Domain
The function given is \( f(x) = 2\tan x \). The domain is \( \left(\frac{\pi}{2}, \frac{3\pi}{2}\right) \). In this interval, the tangent function has a vertical asymptote at \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \).

### Step 2: Analyze the Behavior of the Function
The tangent function, \( \tan x \), is known to be increasing in the interval \( \left(\frac{\pi}{2}, \frac{3\pi}{2}\right) \). Since \( f(x) = 2\tan x \) is just a vertical stretch of the tangent function, it will also be increasing in this interval.

### Step 3: Check for One-One or Many-One
To check if the function is one-one, we can apply the horizontal line test. If any horizontal line intersects the graph of the function more than once, the function is many-one. Since \( f(x) \) is strictly increasing in the given interval, any horizontal line will intersect the graph at most once. Therefore, the function is **one-one**.

### Step 4: Determine If the Function is Onto
Next, we need to check if the function is onto. The codomain is \( \mathbb{R} \). As \( x \) approaches \( \frac{\pi}{2} \) from the right, \( \tan x \) approaches \( +\infty \), and as \( x \) approaches \( \frac{3\pi}{2} \) from the left, \( \tan x \) approaches \( -\infty \). Thus, the range of \( f(x) \) is \( (-\infty, +\infty) \).

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

### Conclusion
The function \( f(x) = 2\tan x \) is:
- **One-One** (injective)
- **Onto** (surjective)