JEE Main 2025 — Functions Question with Solution

$$\text{Step 1: Total Functions with }1\in f(A)$$

Any function $f:A\to B$ where $A=\{1,2,3,4\}$ and $B=\{1,4,9,16\}$ is defined by choosing one of the four elements of $B$ for each element of $A$. Thus, the total number of functions is

$$4^4=256.$$

To count those functions where $1$ appears at least once in the set $f(A)$, we can use the complementary counting method: subtract the functions that never use $1$. If $1$ is excluded, each element of $A$ has only 3 choices (namely, $\{4,9,16\}$), so the number of such functions is

$$3^4=81.$$

Thus, the number of functions such that $1\in f(A)$ is

$$256-81=175.$$

$$\text{Step 2: Counting Many-One Functions}$$

In this context, "many-one functions" are understood to be non-injective functions. Since an injective (one-to-one) function from $A$ to $B$ must be a permutation (because both sets have 4 elements), the number of one-to-one functions is

$$4!=24.$$

It is important to note that every injective function $f:A\to B$ has $f(A)=B$ (a full permutation) which automatically means $1\in f(A)$.

Thus, the number of many-one (non-injective) functions $f:A\to B$ with $1\in f(A)$ is found by subtracting the one-to-one functions from the total functions that include $1$:

$$175-24=151.$$

$$\boxed{151}$$

This detailed explanation shows that the number of many-one functions $f:A\to B$ such that $1\in f(A)$ is indeed $151$.