Sets, Relations & Functions
Functions involving greatest integer
Grade 11
Question:
<p>If \(f(x) = \cos[\pi^2]x + \cos[-\pi^2]x\), which one is correct?</p>
<p>(a) \(f(\pi/2) = -1\)</p>
<p>(b) \(f(\pi) = 1\)</p>
<p>(c) \(f(-\pi) = 0\)</p>
<p>(d) \(f(\pi/4) = 2\)</p>
Step-by-Step Solution
Key Concept: Recognize that [π²] denotes the greatest integer function (floor function), so [π²] = 9 and [-π²] = -10. The function becomes a sum of cosines with integer coefficients that can be simplified using properties of even/odd functions.
<p><strong>Step 1:</strong> Evaluate the greatest integer (floor) function values.</p><p>Since π ≈ 3.14159..., we have π² ≈ 9.8696...</p><p>[π²] = 9 (greatest integer ≤ π²)</p><p>[-π²] = [-9.8696...] = -10 (greatest integer ≤ -π²)</p><p><strong>Step 2:</strong> Substitute into the function.</p><p>f(x) = cos(9x) + cos(-10x)</p><p><strong>Step 3:</strong> Use the even property of cosine: cos(-θ) = cos(θ)</p><p>f(x) = cos(9x) + cos(10x)</p><p><strong>Step 4:</strong> Determine properties of f(x).</p><p>Since both cos(9x) and cos(10x) are even functions, f(x) is an even function: f(-x) = f(x)</p><p>The function is periodic with fundamental period 2π (or a divisor thereof).</p><p>∴ Answer: A</p>
Correct Answer: A