Sets, Relations & Functions
General
Grade 11
Question:
<p>Let \(f(x)=x^2+3x+2\). Which are correct?<br>(A) \(f(|x|)=2\) has 1 solution (B) 3 solutions (C) \(|f(x)|=0.125\) has 4 solutions (D) \(|f(|x|)|=0.125\) has 8 solutions</p>
A,B
<strong>A,C</strong>
C,D
B,D
Step-by-Step Solution
<div class="solution"><p>(A) Let u=|x|\ge0: u^2+3u+2=2 \to u(u+3)=0 \to u=0 \to x=0 only. 1 solution ✓</p><p>(C) |f(x)|=1/8: f(x)=\pm1/8. Each gives \Delta>0 \to 2 roots each \to 4 total ✓</p><p>(D) f(|x|)\ge2 always, so cannot equal 0.125 ✗</p><p><strong>Answer: (A),(C)</strong></p><div class="trap-box"><strong>Trap:</strong> f(|x|)\ge2 kills option (D) instantly.<div class="key-concept"><strong>Key Concept:</strong> u=|x|\ge0 substitution for counting solutions
Correct Answer: A,C