Relations & Functions
Domain of Composite Functions
Grade 12
Question:
<p>The domain of the function <span class='math'>f(x) = \sin^{-1}\left(\frac{1}{2} - \frac{1}{|x-1|} + \sin^{-1}x + \sin x - 1\right)</span> is</p>
<p>(a) <span class='math'>(-\infty, \infty)</span></p>
<p>(b) <span class='math'>(-\infty, -\sqrt{2}] \cup [\sqrt{2}, \infty)</span></p>
<p>(c) <span class='math'>(-\infty, -\sqrt{2}] \cup [\sqrt{2}, \infty) \cup \{0\}</span></p>
<p>(d) None of the above</p>
Step-by-Step Solution
Key Concept: Multiple constraints must be satisfied simultaneously: domain of inverse sine, non-zero denominator, and range of outer inverse sine.
<p>For <span class='math'>\sin^{-1}</span> to be defined, the argument must be in <span class='math'>[-1, 1]</span>. Let <span class='math'>u = \frac{1}{2} - \frac{1}{|x-1|} + \sin^{-1}x + \sin x - 1</span>.</p><p>We need: (1) <span class='math'>|x-1| \neq 0</span>, so <span class='math'>x \neq 1</span>, (2) <span class='math'>\sin^{-1}x</span> requires <span class='math'>x \in [-1, 1]</span>, and (3) <span class='math'>-1 \leq u \leq 1</span>.</p><p>Careful analysis of these constraints together shows the domain is neither of the standard forms given, hence the answer is (d).</p>
Correct Answer: d