Functions
General
Grade 12
Question:
Range of f(x) = \frac{\sec x + \tan x - 1}{\tan x - \sec x + 1}; x \in \left(0, \frac{\pi}{2}\right) is
(0, 1)
(1, \infty)
(-1, 0)
(-\infty, -1)
Step-by-Step Solution
<div class="solution"><p>(A) Let u=|x|≥0: u²+3u+2=2 → u(u+3)=0 → u=0 → x=0 only. 1 solution ✓</p><p>(C) |f(x)|=1/8: f(x)=±1/8. Each gives Δ>0 → 2 roots each → 4 total ✓</p><p>(D) f(|x|)≥2 always, so cannot equal 0.125 ✗</p><p><strong>Answer: (A),(C)</strong></p><div class="trap-box"><strong>Trap:</strong> f(|x|)≥2 kills option (D) instantly.</div><div class="key-concept"><strong>Key Concept:</strong> u=|x|≥0 substitution for counting solutions</div></div>
Correct Answer: B