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RELATIONS AND FUNCTIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsRELATIONS AND FUNCTIONS3 Marks
Question
Show that the Signum Function f: R → R, given by $\text{f(x)}=\begin{cases}1,&\text{if }\text{x }>0\\0,&\text{if }\text{x }=0\\-1,&\text{if }\text{x }<0\end{cases}$ is neither one-one nor onto.
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Answer

f: R → R is given by,
$\text{f(x)}=\begin{cases}1,&\text{if }\text{x }>0\\0,&\text{if }\text{x }=0\\-1,&\text{if }\text{x }<0\end{cases}$
It is seen that f(1) = f(2) = 1, but $1\neq2.$
$\therefore$ f(-1) = f(1), but $-1\neq1.$
$\therefore$ f is not one-one.
Now, as f(x) takes only 3 values (1, 0, or -1) for the element -2 in co-domain R, there does not exist any x in domain R such that f(x) = -2.
$\therefore$ f is not onto.
Hence, the signum function is neither one-one nor onto.

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