JEE Main 2022 — Functions Question with Solution

There are 16 ordered pairs in $S\times S$. We write all these ordered pairs in 4 sets as follows.

$A=\{(1,1)\}$

$B=\{(1,4),(2,4),(3,4)(4,4),(4,3),(4,2),(4,1)\}$

$C=\{(1,3),(2,3),(3,3),(3,2),(3,1)\}$

$D=\{(1,2),(2,2),(2,1)\}$

All elements of set $B$ have image 4 and only element of $A$ has image 1.

All elements of set $C$ have image 3 or 4 and all elements of set $D$ have image 2 or 3 or 4 .

We will solve this question in two cases.

Case I: When no element of set $C$ has image 3.

Number of onto functions $=2$ (when elements of set $D$ have images 2 or 3$)$

Case II: When atleast one element of set $C$ has image 3.

Number of onto functions $=\left(2^3-1\right)(1+2+2)$

$$=35$$

Total number of functions $=37$