Question

If the function $f:R\to R$ be such that $f\left(x\right)=x-\left[x\right]$ denotes the greatest integer less than or equal to $x$, then $f^{-1}\left(x\right)$ is:

• A. $\frac{1}{\left[x-\left[x\right]\right]}$
• B. $\left[x\right]-x$
• C. Not defined
• D. $\left[x\right]+x$

Answer 1

The correct option is C

Not defined

Explanation for the correct option:

Given function,

$\Rightarrow f\left(x\right)=x-\left[x\right]$

$\Rightarrow =\left\{x\right\}$ , i.e, only fraction part of $x$

We need to evaluate its inverse, i.e., $f^{-1}\left(x\right)$

We know the greatest integer function is $\left[x\right]$.

So, for different values of $x$ in $f\left(x\right)$, the same values of the function exists.

For example: If $x=1.5$ , then $f\left(x\right)=0.5$, Similarly, for $x=2.5$, $f\left(x\right)$ will be $0.5$

Thus, $f\left(x\right)$ will always be the same regardless the value of $x$ and hence, we cannot define $f^{-1}\left(x\right)$.

Hence, option(C) i.e. Not defined is the correct answer.