<p>If a function satisfies \(f(x+1) - f(x-1) = 2f(x)\), then period of <i>f</i>(<i>x</i>) can be</p>
Step-by-Step Solution
Key Concept: Solve the functional equation using characteristic equation methods to find valid periods.
<p>The functional equation $f(x+1) - f(x-1) = 2f(x)$ defines a recurrence relation. Rearranging: $f(x+1) - 2f(x) + f(x-1) = 0$. The characteristic equation is $r^2 - 2r + 1 = 0$, giving $(r-1)^2 = 0$, so $r = 1$ (double root). The general solution is $f(x) = (A + Bx)\cos(2πx/T)$ for some period $T$. Testing the options, a period of 8 satisfies the functional equation.</p>
Correct Answer: D