Show that the Modulus Function f : R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative.

Answer

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) = |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.