Question:
If the function $f: R \rightarrow R$ be such that $f(x)=x-[x]$, where $[x]$ denotes the greatest integer less than or equal to $x$, then $f^{-1}(x)$ is
(a) $\frac{1}{x-[x]}$
(b) $[x]-x$
(c) not defined
(d) none of these
Solution:
$f(x)=x-[x]$
We know that the range of $f$ is $[0,1)$.
Co-domain of $f=R$
As range of $f \neq$ Co-domain of $f, f$ is not onto.
$\Rightarrow f$ is not a bijective function.
So, $f^{-1}$ does not exist.
Thus, the answer is (c).
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