Sets, Relations & Functions
Domain of $\sin^{-1}$ of Rational Expression
nta_pyq_2024_apr
Grade 11
Question:
If the domain of the function $f(x)=\sin^{-1}\left(\dfrac{x-1}{2x+3}\right)$ is $\mathbb{R}-(\alpha,\beta)$, then $12\alpha\beta$ is equal to:
Step-by-Step Solution
Key Concept: Need $\left|\frac{x-1}{2x+3}\right|\leq1$ and $2x+3\neq0$. $|x-1|\leq|2x+3|\Rightarrow x\in(-\infty,-4]\cup[-2/3,\infty)$. Excluded interval: $(-4,-2/3)$.
Domain $=\mathbb{R}-(-4,-2/3)$. $\alpha=-4$, $\beta=-2/3$. $12\alpha\beta=32$.
Correct Answer: 1