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Question

Show that the modulus function f:RR 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|={x if x>0x if x<0
It is seen that f(1)=|1|=1,f(1)=|1|=1f(1)=f(1), but 11
f is not one-one
Now, consider 1R.
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.

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