If the range of the function `f(x)= cot^(-1)((x^(2))/(x^(2)+1))` is (a, b), find the value of `((b)/(a)+2)`.

Answer

To find the range of the function \( f(x) = \cot^{-1}\left(\frac{x^2}{x^2 + 1}\right) \) and subsequently the value of \(\frac{b}{a} + 2\), we can follow these steps:

### Step 1: Define the inner function
Let \( g(x) = \frac{x^2}{x^2 + 1} \).

### Step 2: Determine the range of \( g(x) \)
To analyze the range of \( g(x) \), we can rewrite it as:
\[ g(x) = 1 - \frac{1}{x^2 + 1} \]
Here, the denominator \( x^2 + 1 \) is always positive for all real \( x \).

### Step 3: Find the maximum and minimum values of \( g(x) \)
- As \( x \) approaches \( 0 \), \( g(0) = \frac{0^2}{0^2 + 1} = 0 \).
- As \( x \) approaches \( \infty \), \( g(x) \) approaches \( 1 \).

Thus, the range of \( g(x) \) is:
\[ [0, 1) \]

### Step 4: Find the range of \( f(x) \)
Now we need to find the range of \( f(x) = \cot^{-1}(g(x)) \). Since \( g(x) \) takes values in the interval \([0, 1)\):
- When \( g(x) = 0 \), \( f(x) = \cot^{-1}(0) = \frac{\pi}{2} \).
- When \( g(x) \) approaches \( 1 \), \( f(x) = \cot^{-1}(1) = \frac{\pi}{4} \).

Thus, the range of \( f(x) \) is:
\[ \left[\frac{\pi}{4}, \frac{\pi}{2}\right] \]

### Step 5: Identify \( a \) and \( b \)
From the range, we have:
- \( a = \frac{\pi}{4} \)
- \( b = \frac{\pi}{2} \)

### Step 6: Calculate \(\frac{b}{a} + 2\)
Now, we compute:
\[ \frac{b}{a} = \frac{\frac{\pi}{2}}{\frac{\pi}{4}} = \frac{\pi}{2} \cdot \frac{4}{\pi} = 2 \]
Thus,
\[ \frac{b}{a} + 2 = 2 + 2 = 4 \]

### Final Answer
The value of \(\frac{b}{a} + 2\) is \( \boxed{4} \).