Let `f(x)=tan^(-1)((sqrt(1+x^(2))-1)/(x))` then which of the following is correct :

Options

A. f(x) has only one integer in its range
B. Range of f(x) is `(-(pi)/(4), (pi)/(4))-{0}`
C. Range of f(x) is `(-(pi)/(2), (pi)/(2))-{0}`
D. Range of f(x) is `[ -(pi)/(4), (pi)/(4)]-{0}`

Answer

To solve the problem, we need to analyze the function \( f(x) = \tan^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right) \) and find its range.

### Step 1: Rewrite the function
We start with the function:
\[ f(x) = \tan^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right) \]

### Step 2: Substitute \( x = \tan(\theta) \)
Let \( x = \tan(\theta) \). Then, we have:
\[ \sqrt{1+x^2} = \sqrt{1+\tan^2(\theta)} = \sec(\theta) \]
Thus, substituting this into the function gives:
\[ f(x) = \tan^{-1}\left(\frac{\sec(\theta) - 1}{\tan(\theta)}\right) \]

### Step 3: Simplify the expression
Using the identity \( \sec(\theta) = \frac{1}{\cos(\theta)} \) and \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \), we can rewrite the function:
\[ f(x) = \tan^{-1}\left(\frac{\frac{1}{\cos(\theta)} - 1}{\frac{\sin(\theta)}{\cos(\theta)}}\right) = \tan^{-1}\left(\frac{1 - \cos(\theta)}{\sin(\theta)}\right) \]

### Step 4: Use half-angle identities
Using the half-angle identities, we know:
\[ 1 - \cos(\theta) = 2\sin^2\left(\frac{\theta}{2}\right) \quad \text{and} \quad \sin(\theta) = 2\sin\left(\frac{\theta}{2}\right)\cos\left(\frac{\theta}{2}\right) \]
Substituting these into the function gives:
\[ f(x) = \tan^{-1}\left(\frac{2\sin^2\left(\frac{\theta}{2}\right)}{2\sin\left(\frac{\theta}{2}\right)\cos\left(\frac{\theta}{2}\right)}\right) = \tan^{-1}\left(\frac{\sin\left(\frac{\theta}{2}\right)}{\cos\left(\frac{\theta}{2}\right)}\right) \]

### Step 5: Simplify further
This simplifies to:
\[ f(x) = \tan^{-1}\left(\tan\left(\frac{\theta}{2}\right)\right) = \frac{\theta}{2} \]
Since \( \theta = \tan^{-1}(x) \), we have:
\[ f(x) = \frac{1}{2}\tan^{-1}(x) \]

### Step 6: Determine the range of \( f(x) \)
The range of \( \tan^{-1}(x) \) is \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \). Therefore, the range of \( f(x) \) is:
\[ \text{Range of } f(x) = \left(-\frac{\pi}{4}, \frac{\pi}{4}\right) \]

### Step 7: Exclude zero from the range
Since the function \( f(x) \) approaches an indeterminate form when \( x \) approaches 0, we exclude 0 from the range. Thus, the final range of \( f(x) \) is:
\[ \text{Range of } f(x) = \left(-\frac{\pi}{4}, 0\right) \cup \left(0, \frac{\pi}{4}\right) \]

### Conclusion
Thus, the correct option is the one that states the range of \( f(x) \) is \( \left(-\frac{\pi}{4}, 0\right) \cup \left(0, \frac{\pi}{4}\right) \).