Show that the Modulus Function $f : R \rightarrow R$ given by $(x)=|x|$, is neither one - one nor onto, where $|x|$ is $x$, if $x$ is positive or $0$ and $| X |$ is $- x$, if $x$ is negative.
Show that the Modulus Function $f : R \rightarrow R$ given by $(x)=|x|$, is neither one - one nor onto, where $|x|$ is $x$, if $x$ is positive or $0$ and $| X |$ is $- x$, if $x$ is negative.
$f:$ $R \rightarrow R$ is given by $f(x) = |x| = \left\{ {\begin{array}{*{20}{l}}
X&{{\text{ if }}X \geqslant 0} \\
{ - X}&{{\text{ if }}X < 0}
\end{array}} \right.$
It is clear that $f(-1)=|-1|=1$ and $f(1)=|1|=1$
$\therefore f(-1)=f(1),$ but $-1 \neq 1$
$\therefore f$ is not one $-$ one.
Now, consider $-1 \in R$
It is known that $f(x)=|x|$ is always non-negative. Thus, there does not exist any
element $x$ in domain $R$ such that $f(x)=|x|=-1$
$\therefore f$ is not onto.
Hence, the modulus function is neither one-one nor onto.
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