To find the domain of the function \( \frac{f}{g}(x) \), we first need to determine where the function \( g(x) \) is equal to zero since division by zero is undefined.
The function given is:
\[ g(x) = 2x \]
Setting \( g(x) \) to zero:
\[ 2x = 0 \]
Solving for \( x \):
\[ x = 0 \]
This means that \( g(x) \) is zero at \( x = 0 \). For the function \( \frac{f}{g}(x) \) to be defined, \( g(x) \) must be non-zero.
The domain of \( \frac{f}{g}(x) \) will be:
- All real numbers except where \( g(x) \) is zero.
Therefore, the domain of \( \frac{f}{g}(x) \) is:
the set of all real numbers except \( x = 0 \).
Thus, the correct answer is:
the set of all real numbers except \( x = 0 \).