To determine 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 consider where the denominator [tex]\( f(x) \)[/tex] is non-zero. Division by zero is undefined, so the expression [tex]\( \left(\frac{g}{f}\right)(x) \)[/tex] will be undefined wherever [tex]\( f(x) = 0 \)[/tex].
Here are the steps to find the domain:
1. Identify the functions:
- [tex]\( f(x) = x \)[/tex]
- [tex]\( g(x) = 1 \)[/tex]
2. Form the quotient function:
[tex]\[
\left(\frac{g}{f}\right)(x) = \frac{g(x)}{f(x)} = \frac{1}{x}
\][/tex]
3. Determine where the denominator is zero:
- The denominator of the quotient function is [tex]\( f(x) = x \)[/tex].
- Identify the values of [tex]\(x\)[/tex] for which [tex]\(x = 0\)[/tex]:
[tex]\[
f(x) = x = 0 \implies x = 0
\][/tex]
4. Exclude values that make the denominator zero:
- Since [tex]\( f(x) = x \)[/tex] is zero when [tex]\( x = 0 \)[/tex], [tex]\( \left(\frac{g}{f}\right)(x) \)[/tex] is undefined at [tex]\( x = 0 \)[/tex].
5. State the domain:
- The domain of [tex]\( \left(\frac{g}{f}\right)(x) \)[/tex] is all real numbers [tex]\(x\)[/tex] except where the denominator is zero. Therefore, we exclude [tex]\( x = 0 \)[/tex].
Therefore, the domain of the quotient function [tex]\( \left(\frac{1}{x}\right)(x) \)[/tex] is:
[tex]\[
x \neq 0
\][/tex]
Hence, the domain of [tex]\( \left(\frac{g}{f}\right)(x) \)[/tex] is [tex]\( x \neq 0 \)[/tex].
