Question

Show that the Modulus Function f : R → R, given by f(x) = [x], is neither one-one nor onto, where | x | is x, if x is positive and | x | is – x, if x is negative.

Answer 1

f : R → R given by f(x) = | x |.

(a) f(-1) = |-1| = 1, f(1) = |1| = 1.

⇒ – 1 and 1 have the same image.

∴ f is not one-one.

(b) No negative value belonging to codomain of f has any pre-image in its domain.

∴ f is not onto.

Hence, f is neither one-one nor onto.