To determine the domain of the function [tex]\( f(x) = \frac{7}{x} \)[/tex], we have to consider the values of [tex]\( x \)[/tex] that make the function undefined.
1. Identify where the function is undefined:
The function [tex]\( f(x) = \frac{7}{x} \)[/tex] involves division by [tex]\( x \)[/tex]. Division by zero is not allowed in mathematics because it is undefined. Hence,
[tex]\[
x \neq 0
\][/tex]
2. Determine the interval for [tex]\( x \)[/tex]:
Since [tex]\( x \)[/tex] cannot be zero, the function is defined for all real numbers except [tex]\( x = 0 \)[/tex]. This means that the domain includes all real numbers except zero.
3. Express the domain in interval notation:
The domain consists of all numbers less than zero and all numbers greater than zero. In interval notation, this is represented as:
[tex]\[
(-\infty, 0) \cup (0, \infty)
\][/tex]
Therefore, the domain of the function [tex]\( f(x) = \frac{7}{x} \)[/tex] is [tex]\( \boxed{(-\infty, 0) \cup (0, \infty)} \)[/tex].
