To determine the correct domain for the function \( g(x) = \frac{1}{x} - 12 \), we first need to consider the domain of the original function \( f(x) = \frac{1}{x} \).
The function \( f(x) = \frac{1}{x} \) is defined for all real numbers except \( x = 0 \), since division by zero is undefined. Thus, its domain is \( (-\infty, 0) \cup (0, \infty) \).
Now, let's analyze the transformation applied to create \( g(x) \):
- The term \( -12 \) is a constant and does not affect the values of \( x \). The only value of \( x \) that would still make \( g(x) \) undefined is when \( x = 0 \).
So, the domain of \( g(x) = \frac{1}{x} - 12 \) remains the same as that of \( f(x) \):
- The domain of \( g(x) \) is \( (-\infty, 0) \cup (0, \infty) \).
Looking at the answer choices provided:
- A. \( (-\infty < x < 0) \cup (0 < x < \infty) \) is the correct notation for the domain since it correctly describes the values where \( g(x) \) is defined.
Thus, the correct answer is A. \( (-\infty < x < 0) \cup (0 < x < \infty) \).