Understand the function

The function is

Check whether the function is one-one (injective)

A function is one-one if different inputs always give different outputs.

Let us test some values: $f(2)=2$ $f(4)=2$

Since $2 \neq 4$ but $f(2)=f(4)=2,$ the function is not one-one.

Check whether the function is onto (surjective)

A function is onto if every element of the codomain $\mathbb{N}$ is attained as an output.

Now, what values can $f(n)$ take?

Since $f(n)$ is the highest prime factor of $n$, the output is always a prime number.

Hence the range of $f$ is contained in the set of primes: $\text{Range}(f) \subseteq \{2,3,5,7,\dots\}.$

But the codomain is all of $\mathbb{N}$, which contains composite numbers and also $1$.

For example:

• There is no $n$ such that $f(n)=1$.
• There is no $n$ such that $f(n)=4$.
• There is no $n$ such that $f(n)=6$.

Therefore, $f$ is not onto $\mathbb{N}$.

Conclusion

The function is:

• not one-one
• not onto

Therefore the correct option is $\boxed{\text{B: neither one-one nor onto}}.$