<p>Let <i>f</i>(<i>x</i>) be a function on <i>ℝ</i> and <i>f</i>(<i>x</i> − 2) = <i>f</i>(<i>x</i> + 2). If <i>f</i>(<i>x</i>) = 0 has only three real roots in [0, 4] and one of them is 4, then the number of real roots of <i>f</i>(<i>x</i>) = 0 in (−8, 10] is</p>
Step-by-Step Solution
Key Concept: A function with period 4 repeats its zeros every 4 units. Once you identify all zeros in one period [0, 4], you can count how many times the pattern repeats in the given interval.
<p><strong>Step 1:</strong> Given <i>f</i>(<i>x</i> − 2) = <i>f</i>(<i>x</i> + 2), which means <i>f</i>(<i>x</i> − 4) = <i>f</i>(<i>x</i>), so <i>f</i> is periodic with period 4.</p><p><strong>Step 2:</strong> Put <i>x</i> = 0: <i>f</i>(4) = <i>f</i>(0). But <i>f</i>(4) = 0 (given), so <i>f</i>(0) = 0.</p><p><strong>Step 3:</strong> Since <i>f</i> has only 3 roots in [0, 4] and we know <i>x</i> = 0 and <i>x</i> = 4 are roots, the third root in [0, 4] is 2, so <i>f</i>(2) = 0.</p><p><strong>Step 4:</strong> Using periodicity with period 4, the roots repeat every 4 units. In the interval (−8, 10], which has length 18, we can fit 4 complete periods plus part of another. The roots occur at: ..., −6, −4, −2, 0, 2, 4, 6, 8, 10. Therefore, there are <strong>9 roots</strong>.</p><p>∴ Answer is (D).</p>
Correct Answer: D