Question

If \( f(x)=\tan \left(\frac{\pi}{\sqrt{x+1}+4}\right) \) is a real valued function then the range of \( f \) is

• A. \( [-1,1] \)
• B. \( (0,1] \)
• C. \( [-1, \infty) \)
• D. \( \mathbb{R} \)

Hint:

When solving range problems for composite functions \( f(g(h(x))) \), begin with the innermost expression. Find its range first, and then use that result as the input range for the outer functions step by step.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:

To determine the range of the composite function \( f(x) \), we examine how the inner expression \( \frac{\pi}{\sqrt{x+1}+4} \) varies within the allowed values of \( x \), and then observe how those values behave when passed through the tangent function.
Step 2: Key Formula or Approach:

1. For a square root expression \( \sqrt{g(x)} \), the condition \( g(x) \geq 0 \) must hold.
2. The range of the function \( \sqrt{x} \) is \( [0, \infty) \).
3. The tangent function \( \tan \theta \) increases strictly over the interval \( (0, \frac{\pi}{2}) \).
Step 3: Detailed Explanation:

First, identify the domain of \( f(x) \). For the square root to exist: \[ x+1 \geq 0 \implies x \geq -1 \] Next, consider the expression inside the tangent function. Let \[ \theta = \frac{\pi}{\sqrt{x+1}+4} \] Since \( \sqrt{x+1} \ge 0 \) for all \( x \ge -1 \), we have: \[ \sqrt{x+1} + 4 \ge 4 \] Taking the reciprocal reverses the inequality because all quantities are positive: \[ 0 \textless \frac{1}{\sqrt{x+1}+4} \le \frac{1}{4} \] Multiplying through by \( \pi \): \[ 0 \textless \frac{\pi}{\sqrt{x+1}+4} \le \frac{\pi}{4} \] Therefore, the angle \( \theta \) belongs to the interval \( (0, \frac{\pi}{4}] \).
Now evaluate \( f(x) = \tan \theta \) for \( \theta \in (0, \frac{\pi}{4}] \).
Because \( \tan \theta \) is increasing on this interval: \[ \lim_{\theta \to 0^+} \tan \theta \textless f(x) \le \tan \left( \frac{\pi}{4} \right) \] \[ 0 \textless f(x) \le 1 \] Hence, the range of \( f \) is \( (0, 1] \).
Step 4: Final Answer:

The range of the function is \( (0, 1] \).