(i) f : N → N : f(x) = x² is one - one into.

f(x) = x²

⇒y = x²

Since the function f(x) is monotonically increasing from the domain N → N

∴f(x) is one –one

Range of f(x) = (0,∞)≠N(codomain)

∴f(x) is into

∴f : N → N : f(x) = x² is one - one into.

(ii) f : Z → Z : f(x) = x² is many - one into

f(x) = x²

⇒y = x²

in this range the lines cut the curve in 2 equal valued points of y, therefore, the function f(x) = x² is many - one .

Range of f(x) = (0,∞)≠Z(codomain)

∴f(x) is into

∴ f : Z → Z : f(x) = x² is many - one into