Sets, Relations & Functions
Onto Functions — Counting via Sets A and B
nta_pyq_2026_jan
Grade 11
Consider two sets $A=\{x\in\mathbb{Z}:\left|(|x-3|-3)\right|\leq1\}$ and $B=\left\{x\in\mathbb{R}-\{1,2\}:\dfrac{(x-2)(x-4)}{x-1}\log_e(|x-2|)=0\right\}$. Then the number of onto functions $f:A\to B$ is equal to
Step-by-Step Solution
Key Concept: Set $A$: $|(|x-3|-3)|\leq1\Rightarrow2\leq|x-3|\leq4$. For $x\geq3$: $5\leq x\leq7$. For $x<3$: $-1\leq x\leq1$. $A=\{-1,0,1,5,6,7\}$, $|A|=6$. Set $B$: either $\tfrac{(x-2)(x-4)}{x-1}=0$ ($x=4$, since $x\neq1,2$) or $\log_e|x-2|=0$ ($x=3$). $B=\{3,4\}$, $|B|=2$.
$|A|=6$, $|B|=2$. Onto functions $=2^6-2=62$.
Correct Answer: 3