Relations & Functions
Composite Functions and Domain
Grade 12
Question:
<p>If <i>f</i>(<i>x</i>) = 2<i>x</i> + |<i>x</i>|, <i>g</i>(<i>x</i>) = \(\frac{1}{3}\)(2<i>x</i> − |<i>x</i>|), and <i>h</i>(<i>x</i>) = <i>f</i>(<i>g</i>(<i>x</i>)), find the domain of \(\sin^{-1}(h(h(h(...h(x)...))))\) (n times).</p>
<p>(a) [−1, 1]</p>
<p>(b) (−1, −\(\frac{\sqrt{2}}{2}\)] ∪ [\(\frac{\sqrt{2}}{2}\), 1)</p>
<p>(c) (−1, −\(\frac{\sqrt{2}}{2}\))</p>
<p>(d) [\(\frac{\sqrt{2}}{2}\), 1)</p>
Step-by-Step Solution
Key Concept: Determine the range of the composed function g(x) and verify it lies within the domain of the inverse sine function.
<p><strong>Step 1:</strong> For $x \geq 0$: $f(x) = 2x + x = 3x$ and $g(x) = \frac{1}{3}(2x - x) = \frac{x}{3}$</p><p><strong>Step 2:</strong> For $x < 0$: $f(x) = 2x - x = x$ and $g(x) = \frac{1}{3}(2x + x) = x$</p><p><strong>Step 3:</strong> Therefore, $h(x) = f(g(x)) = g(x)$ for appropriate domain.</p><p><strong>Step 4:</strong> Since $g(x) \in [-1, 1]$ for all $x \in [-1, 1]$, and $\sin^{-1}$ requires argument in [−1, 1], the domain is [−1, 1].</p><p>∴ Answer is (a).</p>
Correct Answer: a