Trigonometry & Inverse Trigonometry
Domain of Inverse Trigonometric Functions
Grade 12
Question:
<p>The domain of <strong>f(x) = sin⁻¹(2x)</strong> is</p>
<p>(a) <span style="display:inline-block">\[-\frac{1}{2}, \frac{1}{2}\]</span></p>
<p>(b) <span style="display:inline-block">\[-\frac{1}{4}, \frac{3}{4}\]</span></p>
<p>(c) <span style="display:inline-block">\[-\frac{1}{4}, \frac{1}{4}\]</span></p>
<p>(d) <span style="display:inline-block">\[-\frac{1}{4}, \frac{1}{2}\]</span></p>
Step-by-Step Solution
Key Concept: The inverse sine function is defined only when its argument lies in [-1, 1]. Scale the inequality accordingly.
<p><strong>Step 1:</strong> For \(\sin^{-1}(2x)\) to be defined, we need \(-1 \leq 2x \leq 1\)</p><p><strong>Step 2:</strong> Dividing by 2: \(-\frac{1}{2} \leq x \leq \frac{1}{2}\)</p><p>∴ The domain is \(\left[-\frac{1}{2}, \frac{1}{2}\right]\)</p>
Correct Answer: A