If `f:(-1,1) to B` be a function defined by `f (x) = tan^(-1) ((2x)/(1-x^2))` , then f is both one-one and onto when B is the interval :

Options

A. `[- (pi)/(2) ,(pi)/(2)]`
B. `(-(pi)/(2) ,(pi)/(2))`
C. `[0,(pi)/(2)]`
D. `[0,(pi)/(2))`

Answer

To determine the interval \( B \) for which the function \( f(x) = \tan^{-1} \left( \frac{2x}{1 - x^2} \right) \) is both one-one and onto, we will follow these steps:

### Step 1: Understanding the Function
The function is defined as:
\[ f(x) = \tan^{-1} \left( \frac{2x}{1 - x^2} \right) \]
This function is defined for \( x \) in the interval \( (-1, 1) \).

### Step 2: Substituting \( x \)
To analyze the function, we can substitute \( x \) with \( \tan(\theta) \):
\[ x = \tan(\theta) \]
Then, we can express \( f(x) \) in terms of \( \theta \):
\[ f(x) = \tan^{-1} \left( \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \right) \]

### Step 3: Using the Double Angle Formula
Using the double angle formula for tangent, we have:
\[ \frac{2\tan(\theta)}{1 - \tan^2(\theta)} = \tan(2\theta) \]
Thus, we can simplify \( f(x) \) to:
\[ f(x) = \tan^{-1}(\tan(2\theta)) = 2\theta \]

### Step 4: Finding the Range of \( \theta \)
Since \( x \) varies from \( -1 \) to \( 1 \), we need to determine the corresponding values of \( \theta \):
\[ \theta = \tan^{-1}(x) \]
When \( x = -1 \), \( \theta = -\frac{\pi}{4} \) and when \( x = 1 \), \( \theta = \frac{\pi}{4} \). Therefore, \( \theta \) varies from:
\[ -\frac{\pi}{4} < \theta < \frac{\pi}{4} \]

### Step 5: Finding the Range of \( f(x) \)
Now substituting back for \( f(x) \):
\[ f(x) = 2\theta \]
Thus, the range of \( f(x) \) will be:
\[ 2 \left(-\frac{\pi}{4}\right) < f(x) < 2 \left(\frac{\pi}{4}\right) \]
This simplifies to:
\[ -\frac{\pi}{2} < f(x) < \frac{\pi}{2} \]

### Step 6: Conclusion
The function \( f(x) \) is both one-one and onto when the range \( B \) is:
\[ B = \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \]

### Final Answer
Thus, the interval \( B \) for which \( f \) is both one-one and onto is:
\[ B = \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \]

---