Sets, Relations & Functions
Domain of Function — Multiple Conditions
nta_pyq_2024_jan
Grade 11
Question:
If the domain of the function $f(x)=\cos^{-1}\left(\dfrac{2-|x|}{4}\right)+(\log_e(3-x))^{-1}$ is $[-\alpha,\beta)-\{\gamma\}$, then $\alpha+\beta+\gamma$ is equal to:
Step-by-Step Solution
Key Concept: Condition 1 (for $\cos^{-1}$): $-1\leq\frac{2-|x|}{4}\leq1\Rightarrow|x|\leq6\Rightarrow x\in[-6,6]$. Condition 2 (for $\log^{-1}$): $3-x>0,3-x\neq1\Rightarrow x<3,x\neq2$. Intersect: $x\in[-6,3)-\{2\}$.
Domain: $[-6,3)-\{2\}$. $\alpha=6,\beta=3,\gamma=2$. $\alpha+\beta+\gamma=11$.
Correct Answer: 3