Question

Show that the Modulus Function f: R → R given by, is neither one-one nor onto, where is x, if x is positive or 0 andis − x, if x is negative.

Answer 1

f: R → R is given by,

It is seen that.

∴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) = = −1.

∴ f is not onto.

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