Relations & Functions
Graphical Transformations
Grade 12
Question:
<p><strong>Ex. 10</strong> Let \(g(x) = x - 2k, \forall 2k \leq x < 2(k+1)\) where \(k \in \mathbb{I}\). Which of the following are correct?</p><p>(a) \(g(x) = x + 2, -2 \leq x < 0\)</p><p>(b) \(g(x) = x - 2, 2 \leq x < 4\)</p><p>(c) \(g(x) = x, 0 \leq x < 2\)</p><p>(d) Period of \(g(x)\) is 2</p>
<p>(a) \(g(x) = x + 2, -2 \leq x < 0\)</p>
<p>(b) \(g(x) = x - 2, 2 \leq x < 4\)</p>
<p>(c) \(g(x) = x, 0 \leq x < 2\)</p>
<p>(d) Period of \(g(x)\) is 2</p>
Step-by-Step Solution
Key Concept: Substitute different integer values of k into the piecewise function definition to identify each piece and recognize the periodic pattern.
<p><strong>Solution:</strong> Given, $g(x) = x - 2k, \forall 2k \leq x < 2(k+1), \forall k \in \mathbb{I}$</p><p>For different values of $k$:</p><p>When $k = -1$: $g(x) = x + 2, -2 \leq x < 0$ ✓</p><p>When $k = 0$: $g(x) = x, 0 \leq x < 2$ ✓</p><p>When $k = 1$: $g(x) = x - 2, 2 \leq x < 4$ ✓</p><p>When $k = 2$: $g(x) = x - 4, 4 \leq x < 6$</p><p>Clearly, $g(x)$ is periodic with period 2. ✓</p><p>∴ All options (a), (b), (c), and (d) are correct.</p>
Correct Answer: a,b,c,d