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.