Question

The domain of the function $f\left(x\right)=\ln \left(x-\left[x\right]\right)$ is

• A. $R$
• B. $R-Z$
• C. $\left(0,+\infty \right)$
• D. $Z$

Answer 1

The correct option is A

$R$

Explanation for correct answer Option(s)

Finding the domain of the function $f\left(x\right)$

The given function of $x$ for all real numbers is defined as:

$$\begin{gathered} \Rightarrow x-\left[x\right]>0 \\ \Rightarrow x>\left[x\right] \end{gathered}$$

This is possible only when $x\in R-\left\{z\right\}$

When we put any integer, say $x=1$

$$\begin{gathered} \Rightarrow x>\left[x\right] \\ \Rightarrow 1>\left[1\right] \\ \Rightarrow 1>1 \end{gathered}$$

the condition is false, hence integers cannot be the domain of the given function.

When we put any real number say, $x=1.2$

$$\begin{gathered} \Rightarrow x>\left[x\right] \\ \Rightarrow 1.2>\left[1.2\right] \\ \Rightarrow 1.2>1 \end{gathered}$$

the condition is true.

Hence, the domain of the function is $R$

Therefore, the correct answer is Option (A).