Question

Let \[f:\left( 2,3 \right)\to \left( 0,1 \right)\] be defined by \[f\left( x \right)=x-\left[ x \right]\] then \[{{f}^{-1}}\left( x \right)\] equals
\[\begin{align}
  & \left( \text{A} \right)\text{x-2} \\
 & \left( \text{B} \right)\text{x+1} \\
 & \left( \text{C} \right)\text{x-1} \\
 & \left( \text{D} \right)x+2 \\
\end{align}\]

Answer 1

Hint: These types of problems are pretty straight forward and are very easy to solve. For solving these types of problems, we need to have a clear understanding and a deep knowledge of functions, functional equations and plotting of functions in graphs. For any given function, to find the inverse, we need to find ‘x’ as a function of ‘y’, because, we write
\[\begin{align}
  & y={{f}^{-1}}\left( x \right) \\
 & \Rightarrow x=f\left( y \right) \\
\end{align}\] .
For functions, in which we cannot express ‘x’ as a function of ‘y’, no inverse exists for these types of functions.

Complete step-by-step answer:
Now, we start off the solution to the given problem by expressing the given function, ‘x’ as a function of ‘y’.
In the problem, it is given that the domain of the function \[f\left( x \right)\] , or the acceptable values of ‘x’ for the given function is \[\left( 2,3 \right)\] . Now, from this statement, we can clearly write that,
\[2 < x < 3\]. Thus from the properties of box function, we can clearly write that,
\[\left[ x \right]=2\]
Thus, replacing the value of \[\left[ x \right]\] in our given equation, we can write it in modified form as,
\[f\left( x \right)=x-2\] . Now, writing \[f\left( x \right)\] as \[y\], we can further modify the equation as,
\[y=x-2\]. Now expressing ‘x’ as a function of ‘y’, we write,
\[x=y+2\]
Since here we have replaced \[f\left( x \right)\] as \[y\], we can write,
\[f\left( x \right)=y\] , or in other words, we can write, \[{{f}^{-1}}\left( y \right)=x\] . Now, replacing this value in \[x=y+2\], we get,
\[{{f}^{-1}}\left( y \right)=y+2\].
Now, replacing ‘y’ with ‘x’, we get,
\[{{f}^{-1}}\left( x \right)=x+2\]
This matches with option (D) of our answer.
So, the correct answer is “Option (D)”.

Note: We should remember that box function only gives integer value as output and should not be treated as $x$ . Most of the students commit mistakes in finding out the inverse of a function and so, we should find the inverse carefully step by step, keeping in mind to reverse the domain and range of the function at the end, and otherwise it will lead to wrong answers.