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# Domain of function$f\left( x \right) = \ln \left( x \right)$ where () represents fractional part function A. $R$ B. $R - Z$ C. $(0,\,\infty )$ D. $Z$

Last updated date: 13th Jul 2024
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Hint: A function is a relation which describes that there should be only one output for each input, or we can say that a special kind of relation (a set of ordered pairs), which follows a rule that is every $x$ value must be associated with a $y$value.

Complete step-by-step solution:
We know that, $\ln \left( x \right)$is defined for all places where $x > 0$, that is $x$ should always be positive.
Here, in this question, $f\left( x \right) = \ln \left( x \right)$ (fractional part of $x$). We also know that the range of the fractional part of $x$ is $0 \leqslant (x) < 1$.
But, to define $f\left( x \right)$as a fractional part of $x$, $(x) \ne 0$ , and we also know that $0 \leqslant (x) < 1$ means that domain is where all the real numbers were $(x) = 0$.
And $(x) = 0$when $x$ is some kind of an integer.
Therefore, the total set of all the integers numbers must be removed from real numbers.
So, the domain comes out to be $x = R - Z$.
So, according to the solution, Option B is the right option.

Note: One thing which we should keep in mind is that $\ln \left( x \right)$ function is defined for all $x > 0$( $x$ is always negative, not even equal to 0). The Range of the fractional part of x is always $0 \leqslant (x) < 1$. In Mathematics, domain is a collection of the first values in the order, and range is the collection of the second values.