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If $f:R\rightarrow R$ is defined as $f(x)=[x]^2+[x+1]-3$ where $[x]$ is greatest integer of $x$, then what type of function is $f$?

$\begin{array}{1 1} \text{many to one and onto function.} \\ \text{many to one and into function.} \\\text{one to one and into function.} \\\text{ bijection.} \end{array}$

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  • $[x+n]=[x]+n\:\:if\:\:n\in Z$
$f(x)=[x]^2+[x+1]-3$
$=[x]^2+[x]+1-3$
$=[x]^2+[x]-2$
Factorising the quadratic expression in $[x]$ we get
$f(x)=([x]+2)([x]-1)$
$f(x)=0\Rightarrow\:[x]+2=0\:\:or\:\:[x]-1=0$
$\Rightarrow\:[x]=-2\:\:or\:\:[x]=1$
$\Rightarrow\:x\in[1,2)\:\:or\:\:x\in [-2,-1)$
$\Rightarrow\:f$ is not one to one but is many to one function.
Also since $f(x)=[x]^2+[x]-2$ and $[x]^2,\:[x]$ take only integer values,
$f(x)$ assumes only integer values.i.e., $f(x)\in Z$.
$\Rightarrow\:$ Range of $f$ is Z
But given that $f:R\rightarrow R$
$\Rightarrow\:f$ is not onto function, but is into function.
$\Rightarrow\:f$ is many to one into function.
answered May 28, 2013 by rvidyagovindarajan_1
 

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