Relations & Functions
Even and Odd Functions
Grade 12
Question:
<p>If the function <span style='font-family:monospace'>f(x) = [3.5 + b sin x]</span> (where <span style='font-family:monospace'>[·]</span> denotes the greatest integer function) is an even function, find the complete set of values of <span style='font-family:monospace'>b</span>.</p>
<p>(a) <span style='font-family:monospace'>(-0.5, 0.5)</span></p>
<p>(b) <span style='font-family:monospace'>[-0.5, 0.5]</span></p>
<p>(c) <span style='font-family:monospace'>(0, 1)</span></p>
<p>(d) <span style='font-family:monospace'>[-1, 1]</span></p>
Step-by-Step Solution
Key Concept: For the greatest integer function to be constant (making f even), the argument must stay within a unit interval. The range of b sin x must be restricted.
<p><strong>Step 1:</strong> For <span style='font-family:monospace'>f(x)</span> to be an even function, we need <span style='font-family:monospace'>f(-x) = f(x)</span>.</p><p><strong>Step 2:</strong> We have <span style='font-family:monospace'>f(x) = [3.5 + b sin x]</span>. For this to be even:</p><p><span style='font-family:monospace'>3 ≤ 3.5 + b sin x ≤ 4</span> for all <span style='font-family:monospace'>x ∈ ℝ</span></p><p><strong>Step 3:</strong> This gives us <span style='font-family:monospace'>-0.5 ≤ b ≤ 0.5</span>, so <span style='font-family:monospace'>b ∈ (-0.5, 0.5)</span></p><p>∴ Answer is (a).</p>
Correct Answer: a