Question

The domain of the function $f\left( x \right)=\ln \left( x-\left[ x \right] \right)$ is:
(1) $R$
(2) $R-Z$
(3) $\left( 0,+\infty \right)$
(4) $Z$

Answer 1

Hint: Here in this question we have been asked to find the domain of the given function $f\left( x \right)=\ln \left( x-\left[ x \right] \right)$ . The definition of the domain of a function is the set of all possible inputs for the function. Now for answering this question we will find the possible values of $x$ .

Complete step by step answer:
Now considering from the question we have been asked to find the domain of the given function $f\left( x \right)=\ln \left( x-\left[ x \right] \right)$ .
From the basic concepts of functions, we know that the definition of the domain of a function is the set of all possible inputs for the function.
From the basic concepts of logarithms, we know that the logarithm is defined as ${{\log }_{b}}x=y\Rightarrow {{b}^{y}}=x$ where $x,b>0$ and $b\ne 1$ .
In the given function, the base is $e$ it satisfies the conditions of logarithms and here $x-\left[ x \right]>0$should be valid for satisfying the conditions of logarithms.
We know that $\left[ x \right]$ is the greatest integer function which is defined as the greatest integer value that is less than or equal to $x$ . For example $\left[ -1.5 \right]=-2$ and $\left[ 2.7 \right]=2$ .
We can say that $x-\left[ x \right]=\left\{ \begin{matrix}
   \text{0, if x is an integer} \\
   \text{decimal part of x, if a is a positive non-integer} \\
   \text{(1- decimal part of x), if a is a negative non-integer} \\
\end{matrix} \right\}$.
Since $x-\left[ x \right]>0$ it implies that $x>\left[ x \right]$ is valid when $x$ is not a perfect integer.
Therefore we can conclude that the domain of the given function $f\left( x \right)=\ln \left( x-\left[ x \right] \right)$ is $R-Z$ .

So, the correct answer is “Option 2”.

Note: While answering questions of this type we should be sure with our concepts that we are going to apply in between the steps. If someone had a misconception and considered $\left[ -1.5 \right]=-1$ then they will end up having no conclusion for the question because they will say that $x-\left[ x \right]=\left\{ \begin{matrix}
   \text{0, if x is an integer} \\
   \text{decimal part of x, if a is a positive non-integer} \\
   \text{(- decimal part of x), if a is a negative non-integer} \\
\end{matrix} \right\}$
Which is a wrong statement leading to the conclusion that the domain will be $\left( 0,\infty \right)-Z$ which is not in the options.