Let `f(x) = tan^(-1) (sqrt(x+7)) + sec^(-1)(1/(sqrt(x(x+7)+1)))`, then the range of contains …………. Elements.

Answer

To find the range of the function \( f(x) = \tan^{-1}(\sqrt{x+7}) + \sec^{-1}\left(\frac{1}{\sqrt{x(x+7)+1}}\right) \), we will analyze each component of the function step by step.

### Step 1: Analyze \( \tan^{-1}(\sqrt{x+7}) \)

The function \( \tan^{-1}(y) \) has a range of \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \) for all real values of \( y \).

- The expression inside \( \tan^{-1} \) is \( \sqrt{x+7} \).
- For \( \sqrt{x+7} \) to be defined, we need \( x + 7 \geq 0 \) which implies \( x \geq -7 \).
- As \( x \) approaches \( -7 \), \( \sqrt{x+7} \) approaches \( 0 \), and as \( x \) increases, \( \sqrt{x+7} \) approaches \( +\infty \).
- Therefore, the range of \( \tan^{-1}(\sqrt{x+7}) \) is \( \left(0, \frac{\pi}{2}\right) \).

### Step 2: Analyze \( \sec^{-1}\left(\frac{1}{\sqrt{x(x+7)+1}}\right) \)

The function \( \sec^{-1}(y) \) has a range of \( [0, \frac{\pi}{2}) \cup (\frac{\pi}{2}, \pi] \) for \( |y| \geq 1 \).

- The expression inside \( \sec^{-1} \) is \( \frac{1}{\sqrt{x(x+7)+1}} \).
- For \( \sec^{-1} \) to be defined, we need \( \sqrt{x(x+7)+1} \) to be non-zero, which is always true since \( x(x+7) + 1 \) is always positive for \( x \geq -7 \).
- As \( x \) approaches \( -7 \), \( \sqrt{x(x+7)+1} \) approaches \( 0 \) and thus \( \frac{1}{\sqrt{x(x+7)+1}} \) approaches \( +\infty \).
- As \( x \) increases, \( \sqrt{x(x+7)+1} \) increases, leading \( \frac{1}{\sqrt{x(x+7)+1}} \) to decrease towards \( 0 \).
- Therefore, \( \sec^{-1}\left(\frac{1}{\sqrt{x(x+7)+1}}\right) \) approaches \( 0 \) as \( x \) increases.

### Step 3: Combine the Ranges

Now we combine the ranges of both components:

- The range of \( \tan^{-1}(\sqrt{x+7}) \) is \( (0, \frac{\pi}{2}) \).
- The range of \( \sec^{-1}\left(\frac{1}{\sqrt{x(x+7)+1}}\right) \) is \( [0, \frac{\pi}{2}) \cup (\frac{\pi}{2}, \pi] \).

### Step 4: Final Range of \( f(x) \)

The total range of \( f(x) \) is the union of the two ranges:

- The intersection point is \( 0 \).
- The combined range is \( (0, \frac{\pi}{2}) \cup [0, \frac{\pi}{2}) \).

Thus, the only common point in both ranges is \( 0 \).

### Conclusion

The range of \( f(x) \) contains **1 element**, which is \( 0 \).

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