Show that the modulus function f:R→R given by f(x)=|x|, is neither one-one nor onto, where |x| is x, if x is positive or 0 and |x| is −x, if x is negative.
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Solution
f(x)=|x|={xifx>0−xifx<0 It is seen that f(−1)=|−1|=1,f(1)=|1|=1∴f(−1)=f(1), but −1≠1 ∴f is not one-one Now, consider −1∈R. but 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.