Let `f: N-{1}->N` be defined by, `f(n)=` the highest prime factor of `n` . Show that `f` is neither one-one nor onto. Find the range of `fdot`

Explanation

To show that the function \( f: \mathbb{N} - \{1\} \to \mathbb{N} \) defined by \( f(n) \) as the highest prime factor of \( n \) is neither one-one nor onto, we can follow these steps:

### Step 1: Show that \( f \) is not one-one

To demonstrate that \( f \) is not one-one, we need to find at least two different natural numbers \( n_1 \) and \( n_2 \) such that \( f(n_1) = f(n_2) \).

- Consider \( n_1 = 4 \) and \( n_2 = 8 \).
- The prime factorization of \( 4 \) is \( 2^2 \), so the highest prime factor is \( 2 \).
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