Sets, Relations & Functions
Domain and Range of Ceiling-Minus-x Function
nta_pyq_2023_apr
Grade 11
Question:
Let $A$ and $B$ denote the domain and range respectively of $f(x)=\dfrac{1}{\sqrt{[x]-x}}$, where $[x]$ denotes the smallest integer $\geq x$. Among (S1): $A\cap B=(1,\infty)\setminus\mathbb{N}$ and (S2): $A\cup B=(1,\infty)$,
Only (S2) is true
Only (S1) is true
Neither (S1) nor (S2) is true
Both (S1) and (S2) are true
Step-by-Step Solution
Key Concept: $[x]-x=-(x-[x])=-\{x\}\leq0$, so $\sqrt{[x]-x}$ is not real. Domain $A=\emptyset$.
Domain is empty. Neither statement is true.
Correct Answer: 3