To determine the domain of the function [tex]\( f(x) = \frac{-4}{x} + 1 \)[/tex], we need to identify all values of [tex]\( x \)[/tex] for which the function [tex]\( f(x) \)[/tex] is defined.
### Step-by-Step Solution:
1. Identify Points Where the Function is Undefined:
The function [tex]\(\frac{-4}{x} + 1\)[/tex] will be undefined wherever the denominator [tex]\( x \)[/tex] is zero since division by zero is undefined.
Setting the denominator equal to zero:
[tex]\[
x = 0
\][/tex]
Therefore, the function [tex]\( f(x) \)[/tex] is undefined at [tex]\( x = 0 \)[/tex].
2. Determine the Domain:
The domain of [tex]\( f(x) \)[/tex] includes all real numbers except those for which the function is undefined.
Since the function is undefined at [tex]\( x = 0 \)[/tex], we need to exclude this point from the real number line.
As a result, the domain of the function is all real numbers excluding [tex]\( x = 0 \)[/tex]. In interval notation, this is written as:
[tex]\[
(-\infty, 0) \cup (0, \infty)
\][/tex]
3. Verify the Domain:
We can summarize the domain by saying that [tex]\( x \)[/tex] can take any real number value except [tex]\( x = 0 \)[/tex], confirming our previously written interval notation.
### Conclusion:
By analyzing the points of undefinedness and constructing the correct intervals, we conclude that the domain of [tex]\( f(x) = \frac{-4}{x} + 1 \)[/tex] is:
[tex]\[
(-\infty, 0) \cup (0, \infty)
\][/tex]
### Correct Answer:
The correct answer is:
[tex]\[ \boxed{D. \, (-\infty, 0) \cup (0, \infty)} \][/tex]
