Q. 5 Show that the Signum Function f : R \rightarrow R, given by

                            f (x) = \left\{\begin{matrix} 1 & if\;x>0 \\ 0& if\;x=0 \\ -1& if\;x<0 \end{matrix}\right.

is neither one-one nor onto.

Answers (1)

f : R \rightarrow R  is given by 

f (x) = \left\{\begin{matrix} 1 & if\;x>0 \\ 0& if\;x=0 \\ -1& if\;x<0 \end{matrix}\right.

As we can see    f(1)=f(2)=1   ,  but 1\neq 2

So it is not one-one.

Now, f(x) takes only 3 values (1,0,-1) for the element -3 in codomain R ,there does not exists x in domain R such that f(x)= -3.

So it is not onto.

Hence, signum function is neither one-one nor onto.

 

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