Sets, Relations & Functions
Domain of Function — Intersection of Conditions
nta_pyq_2024_jan
Grade 11
Question:
If the domain of the function $f(x)=\dfrac{\sqrt{x^2-25}}{(4-x^2)}+\log_{10}(x^2+2x-15)$ is $(-\infty,\alpha)\cup[\beta,\infty)$, then $\alpha^2+\beta^3$ is equal to:
Step-by-Step Solution
Key Concept: Find each condition: $x^2-25\geq0\Rightarrow|x|\geq5$; $4-x^2\neq0\Rightarrow x\neq\pm2$; $x^2+2x-15>0\Rightarrow(x+5)(x-3)>0$. Intersect all conditions.
Conditions: $x\leq-5$ or $x\geq5$; $x\neq\pm2$; $x<-5$ or $x>3$. Domain: $(-\infty,-5)\cup[5,\infty)$. $\alpha=-5,\beta=5$. $\alpha^2+\beta^3=25+125=150$.
Correct Answer: 3