Which of the following functions is an odd function?
|x - 2| + (x + 2) \text{sgn}(x + 2)
\frac{1}{x(e - 1)} + \frac{1}{2x}
\log(\sin(x + \sqrt{1 + \sin^2 x}))
e^{4x} (2^{x - 1})^4
Step-by-Step Solution
<div class="solution"><p>Step 1: Two symmetries → f(t)=f(t-16) → period 16.</p><p>Step 2: f(0)=5 → f(16k)=5 for k=0,1,...,10: that's 11 values (0,16,...,160).</p><p>Step 3: Symmetry about x=2 gives f(0)=f(4)=5 → f(4+16k)=5 for k=0,...,10: another 11 values (4,20,...,164).</p><p>Total minimum = 22.</p><p><strong>Answer: (A) 22</strong></p><div class="trap-box"><strong>Trap:</strong> Symmetry about x=2 forces a second AP. Don't count only 16k.</div><div class="key-concept"><strong>Key Concept:</strong> Two reflections → period; symmetry centre shifts by T/2</div></div>
Correct Answer: C