Consider the identity function 1_N : N → N defined as l_N(x) = x ∀ x ∈ N. Show that although I_N is onto but I_N + I_N : N → N defined as (I_N + I_N) (x) = I_N(x) + I_N(x) = x + x = 2 x is not onto.

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.

(i)  f : R → R defined by f (x) = 3 – 4 x
(ii) f : R → R defined by f (x) = 1 + x².

Let f : N – {1} → N defined by f (n) = the highest prime factor of n. Show that f is neither one-to-one nor onto. Find the range of f.