Trigonometry & Inverse Trigonometry
Domain of sin⁻¹(1/(x²−2x−2)) — Sum of Endpoints
nta_pyq_2026_jan
Grade 12
Question:
If the domain of the function $f(x)=\sin^{-1}\!\left(\dfrac{1}{x^2-2x-2}\right)$ is $(-\infty,\alpha]\cup[\beta,\gamma]\cup[\delta,\infty)$, then $\alpha+\beta+\gamma+\delta$ is equal to
Step-by-Step Solution
Key Concept: Need $\left|\tfrac{1}{x^2-2x-2}\right|\leq1$, i.e., $|x^2-2x-2|\geq1$. Case 1: $x^2-2x-2\geq1\Rightarrow x^2-2x-3\geq0\Rightarrow(x-3)(x+1)\geq0\Rightarrow x\leq-1$ or $x\geq3$. Case 2: $x^2-2x-2\leq-1\Rightarrow(x-1)^2\leq2\Rightarrow1-\sqrt{2}\leq x\leq1+\sqrt{2}$.
$\alpha+\beta+\gamma+\delta=4$.
Correct Answer: 3