Sets, Relations & Functions
General
Grade 11
Question:
<p>Let <span class="math-inline">\(f(x)=x^2+3x+2\)</span>. Which are correct?<br>(A) <span class="math-inline">\(f(|x|)=2\)</span> has 1 solution (B) 3 solutions (C) <span class="math-inline">\(|f(x)|=0.125\)</span> has 4 solutions (D) <span class="math-inline">\(|f(|x|)|=0.125\)</span> has 8 solutions</p>
Step-by-Step Solution
Key Concept: General
<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: A,C