If `f(x)=tan^(-1)sqrt(x^(2)+4x)` `+sin^(-1)sqrt(x^(2)+4x+1)`

Options

A. domain of `f(x)` contains 3 integers only
B. range of `f(x)` has two elements only
C. `f(x)` is a constant function `Aax in R`
D. `f(x)` contains only two elements in its domain

Answer

To solve the problem, we need to analyze the function \( f(x) = \tan^{-1}(\sqrt{x^2 + 4x}) + \sin^{-1}(\sqrt{x^2 + 4x + 1}) \) and determine its domain and range.

### Step 1: Determine the domain of \( f(x) \)

1. **Identify the conditions for the inverse sine function**:
The function \( \sin^{-1}(y) \) is defined for \( y \) in the range \([-1, 1]\). Therefore, we need:
\[ \sqrt{x^2 + 4x + 1} \leq 1 \]

2. **Square both sides**:
Squaring both sides gives:
\[ x^2 + 4x + 1 \leq 1 \]
Simplifying this, we get:
\[ x^2 + 4x \leq 0 \]

3. **Factor the quadratic**:
Factoring the left side:
\[ x(x + 4) \leq 0 \]

4. **Find the critical points**:
The critical points are \( x = 0 \) and \( x = -4 \).

5. **Test intervals**:
We test intervals around the critical points:
- For \( x < -4 \), say \( x = -5 \): \( (-5)(-1) > 0 \) (not in the domain)
- For \( -4 < x < 0 \), say \( x = -2 \): \( (-2)(2) < 0 \) (in the domain)
- For \( x > 0 \), say \( x = 1 \): \( (1)(5) > 0 \) (not in the domain)

Thus, the domain of \( f(x) \) is \( [-4, 0] \).

### Step 2: Check the condition for the inverse tangent function

1. **Condition for the inverse tangent function**:
The function \( \tan^{-1}(y) \) is defined for all real \( y \). Therefore, we need:
\[ \sqrt{x^2 + 4x} \geq 0 \]
This is satisfied for all \( x \) in the domain \( [-4, 0] \).

### Step 3: Calculate the range of \( f(x) \)

1. **Evaluate \( f(x) \) at the endpoints of the domain**:
- For \( x = -4 \):
\[ f(-4) = \tan^{-1}(\sqrt{(-4)^2 + 4(-4)}) + \sin^{-1}(\sqrt{(-4)^2 + 4(-4) + 1}) = \tan^{-1}(0) + \sin^{-1}(1) = 0 + \frac{\pi}{2} = \frac{\pi}{2} \]
- For \( x = 0 \):
\[ f(0) = \tan^{-1}(\sqrt{0^2 + 4(0)}) + \sin^{-1}(\sqrt{0^2 + 4(0) + 1}) = \tan^{-1}(0) + \sin^{-1}(1) = 0 + \frac{\pi}{2} = \frac{\pi}{2} \]

2. **Conclusion on the range**:
Since \( f(-4) = f(0) = \frac{\pi}{2} \), the range of \( f(x) \) is a single value:
\[ \text{Range} = \left\{ \frac{\pi}{2} \right\} \]

### Final Conclusion
- The domain of \( f(x) \) contains only 2 elements: \( -4 \) and \( 0 \).
- The range of \( f(x) \) has only 1 element: \( \frac{\pi}{2} \).
- Therefore, the correct option is that \( f(x) \) is a constant function.