JEE Mathematics 2024 Question with Solution

Given: The function is $f: \mathbb{N} - \{1\} \to \mathbb{N}$ where $f(n)$ is the highest prime factor of $n$.

Find: Whether the function is one-one, onto, both, or neither.

To test whether $f$ is one-one, consider two different inputs:

Both $6$ and $9$ have the same image $3$. Therefore, different elements of the domain map to the same value, so $f$ is not one-one.

To test whether $f$ is onto, note that the codomain is $\mathbb{N}$, but the output of $f(n)$ is always a prime number because it is the highest prime factor of $n$.

Hence a natural number such as $4$ cannot be the image of any $n$, since $4$ is not prime. So every element of $\mathbb{N}$ is not attained. Therefore, $f$ is not onto.

Therefore, the function is neither one-one nor onto. The correct option is D.

Given: $f(n)$ returns the highest prime factor of $n$ for $n \in \mathbb{N} - \{1\}$.

Find: The nature of the function.

Examples from the definition:

These show that the output is always a prime number.

For injectivity, if the function were one-one, distinct inputs would have distinct outputs. But

so $f$ is not injective.

For surjectivity, every element of $\mathbb{N}$ should appear as an output. Since only prime numbers can occur as values of $f(n)$, composite numbers such as $4$ are never attained. Thus $f$ is not surjective.

So the function is neither one-one nor onto, hence the correct option is D.