Show that the Signum Function f : R R, given by f(x)= l 1, if x>0… - Vidyadip

Question

Show that the Signum Function $f : R \rightarrow R$, given by $f(x)=\left\{\begin{array}{l}1, \text { if } x>0 \\ 0, \text { if } x=0 \\ 1, \text { if } x<0\end{array}\right.$ is neither one$-$one nor onto.

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Answer

Signum Function $f : R \rightarrow R, $ given by $f(x) = \left\{ {\begin{array}{*{20}{c}} {1,\;if\;x > 0} \\ {0,\;if\;x = 0} \\ { - 1,\;if\;x < 0} \end{array}} \right.$
$f(1) = f(2) = 1$
Two distinct elements have same image.
$\therefore f $ is not one$-$one.
Except $-1, 0, 1$ no other members of co $-$ domain of f has any pre $-$ image its domain.
$\therefore f$ is not onto.
Therefore $, f$ is neither one $-$ one nor onto.

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