Sets, Relations & Functions
Grade 11
Question:
<p>Let \(f:[1,\infty)\to[3,\infty)\), \(f(x)=(\log_2 x)^2+2\log_2 x+3\). Which are correct?</p>
A,C
<strong>A,D</strong>
B,C
B,D
Step-by-Step Solution
<div class="solution"><p>Let u=log_2x\ge0. Then y=(u+1)^2+2, so u+1=\sqrt(y-2) (positive branch). Thus x=2^{-1+\sqrt(y-2)}.</p><p>Range of f is [3,\infty) \to domain of f⁻^1 is [3,\infty).</p><p><strong>Answer: (A),(D)</strong></p><div class="trap-box"><strong>Trap:</strong> Negative root is invalid since log_2x\ge0 on given domain.<div class="key-concept"><strong>Key Concept:</strong> Log-inside-quadratic \to substitute u=logₐx first
Correct Answer: A,D