The domain of the function
$f(x) = \dfrac{1}{{\sqrt {\left| x \right| - x} }}$ is
  A.( - \infty ,\infty ) \\
  B.(0,\infty ) \\
  C.( - \infty ,0) \\
  D.( - \infty , - \infty ) - \{ 0\} \\

Answer Verified Verified
Hint:This is a simple question based on function and its domain set and codomain set. As we know that domain is the set of all the values that go into a function. We will find the set of possible values which will satisfy the given function.

Complete step-by-step answer:
Given function is:
$f(x) = \dfrac{1}{{\sqrt {\left| x \right| - x} }}$
It is a modulus function in its denominator under the square root.
This function will be defined only for the non-zero value of the denominator.
It means, $
  \sqrt {\left| x \right| - x} > 0 \\
Which further shows that, $\left| x \right| - x > 0$
$ \Rightarrow \left| x \right| > x$
The above expression will be valid only for the negative values of variable x. Also for the positive values it will be invalid.
Thus, the domain of function will be $( - \infty ,0)$. Here the set must be open on both sides.

So, the correct answer is “Option C”.

Note:While introducing the functions in mathematics, we have to define its formula with the possible set of values for domain and range values. The function is for representing the relationship between domain and codomain sets. This relation will be valid for some specific domain. This domain will be the set of valid points. This function is also termed as mapping. The domain of a function may be represented along the x axis and its range values can be represented along the y-axis.