To determine the domain of the function [tex]\(\left(\frac{g}{f}\right)(x)\)[/tex], follow these steps:
1. Understand the Functions:
- [tex]\( f(x) = x \)[/tex]
- [tex]\( g(x) = 1 \)[/tex]
2. Form the Composite Function [tex]\(\left(\frac{g}{f}\right)(x)\)[/tex]:
[tex]\[
\left(\frac{g}{f}\right)(x) = \frac{g(x)}{f(x)} = \frac{1}{x}
\][/tex]
3. Identify Restrictions on the Domain:
- The expression [tex]\(\frac{1}{x}\)[/tex] is a fraction. For a fraction to be defined, its denominator cannot be zero.
- Therefore, [tex]\( f(x) = x \)[/tex] must not equal zero because division by zero is undefined.
4. State the Domain:
- The value [tex]\( x = 0 \)[/tex] makes the denominator zero, which is not allowed.
- Hence, [tex]\( x \)[/tex] can be any real number except zero.
So, the domain of [tex]\(\left(\frac{g}{f}\right)(x)\)[/tex] is:
[tex]\[
\boxed{x \neq 0}
\][/tex]
