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1.Relation and Function
medium
सिद्ध कीजिए कि $f(x)=|x|$ द्वारा प्रद्त मापांक फलन $f: R \rightarrow R$, न तो एकेकी है और न आच्छादक है, जहाँ $|x|$ बराबर $x$, यदि $x$ धन या शून्य है तथा $|x|$ बराबर $-x$, यदि $x$ रुण है।
Option A
Option B
Option C
Option D
Solution
$f:$ $R \rightarrow R$ 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.
Standard 12
Mathematics
