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Relations and Functions — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsRelations and Functions2 Marks
Question
Show that the Signum Function f : R $\rightarrow$ R, given by $f(x) = \left\{ {\begin{array}{*{20}{c}} {1,\;if\;x > 0} \\ {0,\;if\;x = 0} \\ { 1,\;if\;x < 0} \end{array}} \right.$ is neither one-one nor onto.
✓

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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