To find the domain of the quotient of functions [tex]\( \left(\frac{g}{f}\right)(x) \)[/tex], where [tex]\( f(x) = x \)[/tex] and [tex]\( g(x) = 1 \)[/tex], we need to ensure that the denominator [tex]\( f(x) \)[/tex] is not equal to zero.
Given:
- [tex]\( f(x) = x \)[/tex]
- [tex]\( g(x) = 1 \)[/tex]
The quotient function is:
[tex]\[ \left(\frac{g}{f}\right)(x) = \frac{1}{x} \][/tex]
For the quotient [tex]\( \frac{1}{x} \)[/tex] to be defined, the denominator [tex]\( x \)[/tex] must not be zero. Therefore, we exclude [tex]\( x = 0 \)[/tex] from the domain of the function.
Thus, the domain of [tex]\( \left(\frac{g}{f}\right)(x) \)[/tex] is all real numbers except zero. In mathematical notation, this can be written as:
[tex]\[ x \neq 0 \][/tex]
Therefore, the correct answer is:
[tex]\[ x \neq 0 \][/tex]
