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1.Relation and Function
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Let $R$ be the set of real numbers and $f: R \rightarrow R$ be defined by $f(x)=\frac{\{x\}}{1+[x]^2}$, where $[x]$ is the greatest integer less than or equal to $x$, and $\left\{x{\}}=x-[x]\right.$. Which of the following statements are true?
$I.$ The range of $f$ is a closed interval.
$II.$ $f$ is continuous on $R$.
$III.$ $f$ is one-one on $R$
A
$I$ only
B
$II$ only
C
$III$ only
D
None of $I$,$II$ and $III$
(KVPY-2017)
Solution
(d)
We have,
$f(x) =\frac{\{x\}}{1+[x]^2}$
$\Rightarrow \quad f(x) =\frac{x-[x]}{1+[x]^2}$
Range of $f(x)=[0,1)$.
$\therefore$ Range of $f$ is semi-closed interval $f$ is discontinuous on integer.
Clearly, $f$ is not one-one function.
$\therefore$ Option $(d)$ is correct.
Standard 12
Mathematics
