Question

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. \[Z\]

Answer 1

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.