Sets, Relations & Functions
General
Grade None
Question:
Let f : \(\mathbb{R} \to \mathbb{R}\) be defined by \(f(x) = \frac{x}{1 + x^2}, x \in \mathbb{R}\). Then the range of f is :
(-1, 1) - {0}
[\(\frac{1}{2}\), \(\frac{1}{2}\)]
R - [\(\frac{1}{2}\), \(\frac{1}{2}\)]
R - [-1, 1]
Step-by-Step Solution
<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