Question

Show that the Modulus Function f: R$\to$R given by f(x) = IxI, is neither one-one nor onto, where IxI is x, if x is positive or 0 and IxI is −x, if x is negative.

Answer 1

f: R$\to$R is given by,

$f(x) = |x|= \begin{cases} x, & \quad \text{if } x {\geq0}\\\ -x, & \quad \text{if } n {<0} \end{cases}$

It is seen that f(-1)=I-1I=1, f(1)=I1I=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) = = −1.
∴ f is not onto.

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