Let f:N-(1)-N be defined by f(x) = the highest prime factor of x. Show that f is neithet one-one nor onto

Dear Student,

We have the function f:$\mathrm{\mathbb{N}}-\left\{1\right\}\to \mathrm{\mathbb{N}}$ given by
f(x) = highest prime factor of f
Since f(2)=2=f(4), so f is not one-one.

Now, range of f is $\mathrm{\mathbb{N}}$, but only prime numbers have a pre-image under f, 
i.e there is no element of $\mathrm{\mathbb{N}}-\left\{1\right\}$ that is mapped under f to composite natural numbers.
In particular, 4 doesn't have a pre-image under f in $\mathrm{\mathbb{N}}-\left\{1\right\}$ .
Hence, f is not onto.

Regards