Question

Prove that the function f : R→R given by f(x) = |x| is neither one-one nor onto, where || is x, if x is positive or 0 and |x|  is − x, if x is negative.

Answer 1

∴f(−1) = f(1), but −1 ≠ 1.

∴ f is not one-one.

Now, consider −1 ∈ R.

It is known that f(x) = 
|| is always non-negative. Thus, there does not exist any element x in domain R such that f(x) =|x| = −1.

∴ f is not onto.

Hence, the modulus function is neither one-one nor onto.