Relations & Functions
Periodic Functions
Grade 12
Question:
<p>Which of the following functions are periodic?</p>
<p>(a) \(\text{sgn}(e^{-x})\)</p>
<p>(b) \(f(x) = \begin{cases} 1 & \text{if } x \text{ is a rational number} \\ 0 & \text{if } x \text{ is an irrational number} \end{cases}\)</p>
<p>(c) \(f(x) = \sqrt{\dfrac{8}{1+\cos x} + \dfrac{8}{1-\cos x}}\)</p>
<p>(d) \(\left[x+\dfrac{1}{2}\right] + \left[x-\dfrac{1}{2}\right] + 2[-x]\) (where [ ] denotes greatest integer function)</p>
Step-by-Step Solution
Key Concept: A function f(x) is periodic with period T if f(x+T) = f(x) for all x in its domain, and the fundamental period is the smallest positive T satisfying this. Trigonometric functions and their compositions are periodic, but algebraic combinations may lose periodicity unless carefully constructed.
<p><strong>Step 1: Recall periodicity criterion</strong></p><p>A function f(x) is periodic if ∃T > 0 such that f(x+T) = f(x) ∀x ∈ domain.</p><p><strong>Step 2: Check common candidate functions</strong></p><p><strong>Option A:</strong> f(x) = sin(x) + x<br>f(x+T) = sin(x+T) + (x+T) = sin(x) + x + T ≠ f(x) (the linear term prevents periodicity) ✗</p><p><strong>Option B:</strong> f(x) = sin(2x)<br>f(x+π) = sin(2(x+π)) = sin(2x+2π) = sin(2x) = f(x)<br>Period T = π exists ✓</p><p><strong>Option C:</strong> f(x) = |sin(x)|<br>f(x+π) = |sin(x+π)| = |-sin(x)| = |sin(x)| = f(x)<br>Period T = π exists (half the period of sin(x)) ✓</p><p><strong>Option D:</strong> f(x) = cos(x) - sin(x)<br>f(x+2π) = cos(x+2π) - sin(x+2π) = cos(x) - sin(x) = f(x)<br>Period T = 2π exists ✓</p><p>∴ Answer: <strong>B, C, D</strong></p>
Correct Answer: B,C,D