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 or 0 and |x| is –x, if x is negative.

f(x) = |x|

Let x₁ = 1 and x₂ = -1 be two real numbers

f(x₁) = f(1) = |1| = 1;

f(x₂) = f(-1) = |-1| = 1

∴  f(x₁) = f(x₂) for x₁ ≠ x₂

∴  f : R → R  is not one-one

Also, let y = –1  R

∴ f(x) = -1 ⇒ |x| = -1

which is not possible as |x| is always positive.

∴ f : R → R is not onto.