Sets, Relations & Functions
General
Grade 11
Question:
<p><span class="math-inline">\(f(x)=|x|(x-\sin x)\)</span>. Which is true?</p>
One-one not onto
Onto not one-one
<strong>Bijective</strong>
Neither
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>x-sin x is odd increasing. For x>0: f'(x)=(x-sin x)+x(1-cos x)>0 → strictly increasing. Odd, range=ℝ → bijective.</p><p><strong>Answer: (C) bijective</strong></p><div class="trap-box"><strong>Trap:</strong> The |x| factor preserves oddness since it multiplies another odd factor.</div><div class="key-concept"><strong>Key Concept:</strong> Odd × odd = even? No — |x| is even, (x-sin x) is odd, product is odd</div></div>
Correct Answer: 3