Question

The range of \( f(x) = \sec\left( \frac{\pi}{4} \cos^2 x \right), \; -\infty<x<\infty \) is

• A. \( [1, \sqrt{2}] \)
• B. \( [1, \infty) \)
• C. \( [-\sqrt{2}, -1] \cup [1, \sqrt{2}] \)
• D. \( [-\infty, -1] \cup [1, \infty) \)

Hint:

For composite functions, always find the range from inside to outside step-by-step.

Answer 1

The Correct Option is A

Concept: Find range step-by-step by analyzing inner function first, then applying outer function.

Step 1: Range of \( \cos^2 x \). \[ 0 \le \cos^2 x \le 1 \]

Step 2: Multiply by \( \frac{\pi}{4} \): \[ 0 \le \frac{\pi}{4}\cos^2 x \le \frac{\pi}{4} \]

Step 3: Apply secant function: \[ \sec \theta = \frac{1}{\cos \theta}, \quad \theta \in \left[0, \frac{\pi}{4}\right] \] \[ \cos \theta \in \left[\frac{1}{\sqrt{2}}, 1\right] \] \[ \Rightarrow \sec \theta \in [1, \sqrt{2}] \] Final Answer: \[ [1, \sqrt{2}] \]