Sets, Relations & Functions
General
Grade 11
Question:
<p>If <span class="math-inline">\(f\left(x+\frac{1}{2}\right)+f\left(x-\frac{1}{2}\right)=f(x)\)</span> for all <span class="math-inline">\(x\in\mathbb{R}\)</span>, the period is greater than:</p>
A,B
<strong>A,B,C</strong>
A,B,C,D
B,C
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>Shift relation by ±1/2: combine to get f(x+1)+f(x)+f(x-1)=0.</p><p>This standard recurrence gives f(x+3)=f(x), so period=3.</p><p>Greater than 1✓, 3/2✓, 2✓ but NOT greater than 3.</p><p><strong>Answer: (A),(B),(C)</strong></p><div class="trap-box"><strong>Trap:</strong> Stop at the half-step relation. Full period visible only after converting to unit-step recurrence.</div><div class="key-concept"><strong>Key Concept:</strong> Half-step functional equation → unit recurrence → period 3</div></div>
Correct Answer: A,B,C