Sets, Relations & Functions
General
Grade 11
Question:
<p><span class="math-inline">\(f:\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\to\mathbb{R}\)</span>, <span class="math-inline">\(f(x)=[\log(\sec x+\tan x)]^3\)</span>. Which are true?</p>
A,B,C,D
<strong>A,B,C</strong>
A,C
B,C,D
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>Inner log is odd (using (sec x+tan x)(sec x-tan x)=1). Cubing preserves oddness → f odd. Strictly increasing → one-one. Range is all ℝ → onto.</p><p><strong>Answer: (A),(B),(C)</strong></p><div class="trap-box"><strong>Trap:</strong> The reciprocal identity immediately reveals the odd nature of log(sec x+tan x).</div><div class="key-concept"><strong>Key Concept:</strong> Odd + strictly monotone + limits ±∞ → bijective onto ℝ</div></div>
Correct Answer: A,B,C