<p>Let <span class="math-inline">\(f:(1,3)\to\mathbb{R}\)</span>, <span class="math-inline">\(f(x)=\dfrac{x[x]}{1+x^2}\)</span>. Find the range of <span class="math-inline">\(f\)</span>.</p>
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>For x∈(1,2): [x]=1, f=x/(1+x²), decreasing → range (2/5,1/2). For x∈[2,3): [x]=2, f=2x/(1+x²), decreasing → range (3/5,4/5]. Union=(2/5,1/2)∪(3/5,4/5].</p><div class="trap-box"><strong>Trap:</strong> At x=2, [x]=2 so 4/5 is included.</div><div class="key-concept"><strong>Key Concept:</strong> Floor function range — split domain where floor is constant</div></div>
Correct Answer: (2/5,1/2)∪(3/5,4/5]