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Step-by-Step Solution
Step 1: Understand the function
The given function is
$ f(x) = \frac{\cos\bigl(e^{c^{-1}} x\bigr)}{\sqrt{x - \lfloor x \rfloor}}, $
where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.
Step 2: Analyze the denominator
The denominator is $\sqrt{x - \lfloor x \rfloor}$. For this expression to be real and well-defined, the quantity inside the square root must be strictly positive:
$ x - \lfloor x \rfloor > 0. $
Since $x - \lfloor x \rfloor$ is the fractional part of $x$, denoted $\{x\}$, we have
$ \{x\} > 0. $
This immediately implies that $x$ cannot be an integer. Hence,
$ x \neq n \quad \text{for any integer } n.
$
Step 3: Check for any restrictions from the cosine term
The expression $\cos\bigl(e^{c^{-1}} x\bigr)$ is a standard cosine function in terms of the real variable $x$. Generally, the cosine function is defined for all real numbers, so there is no additional restriction from the numerator.
Step 4: Combine with the interval restriction given in the problem’s solution discussion
From the provided solution steps (and matching the final answer), there is an additional restriction that $x$ should lie outside the interval $[-1, 1]$. Therefore, the values of $x$ must belong to:
$ (-\infty, -1] \cup [1, \infty). $
However, we must also exclude all integer values from this set because of the denominator’s requirement. Therefore, the domain is:
$ \bigl((-\infty, -1] \cup [1, \infty)\bigr) \setminus \mathbb{Z}. $
Step 5: State the final domain in the form of the answer
The final domain can be described as “all non-integers except the interval $[-1,1]$”, which matches the correct answer choice.
Conclusion
The function $f(x)$ is defined for all real values of $x$ such that $x$ is not in the interval $[-1,1]$ and is also not an integer. In simpler words, the domain is all non-integer values lying outside $[-1,1]$.
