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 find the range of the composite function f(x), we need to analyze the variation of the inner expression π / (√(x + 1) + 4) over the valid domain of x, and then map those values through the tangent function.

Step 2: Key Formula or Approach:

1. The domain of a square root expression requires the quantity inside the root to be greater than or equal to 0.
2. The range of √x is [0, ∞).
3. The function tan θ is strictly increasing on the interval (0, π/2).

Step 3: Detailed Explanation:

First, determine the domain of f(x). For the square root to be defined:

x + 1 ≥ 0 ⟹ x ≥ -1

Next, analyze the range of the argument inside the tangent function. Let the argument be θ:

θ = π / (√(x + 1) + 4)

Since √(x + 1) ≥ 0 for all x ≥ -1:

√(x + 1) + 4 ≥ 4

Taking the reciprocal reverses the inequality because all values are positive:

0 < 1 / (√(x + 1) + 4) ≤ 1/4

Multiplying by π:

0 < π / (√(x + 1) + 4) ≤ π/4

So, the argument θ lies in the interval (0, π/4].

Now, evaluate f(x) = tan θ for θ ∈ (0, π/4].

Since tan θ is an increasing function in this interval:

limθ→0+ tan θ < f(x) ≤ tan(π/4)

0 < f(x) ≤ 1

Thus, the range of f is (0, 1].

Step 4: Final Answer:

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