To determine the domain of [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] where [tex]\(f(x) = x\)[/tex] and [tex]\(g(x) = 1\)[/tex], we need to analyze when the function [tex]\(\frac{f(x)}{g(x)} = \frac{x}{1}\)[/tex] is defined.
1. Identify the denominator and its constraints:
- Here, [tex]\(g(x) = 1\)[/tex].
- The denominator of a fraction must not be zero to ensure the fraction is defined.
2. Check when the denominator is zero:
- [tex]\(g(x) = 1\)[/tex] does not equal zero for any real number [tex]\(x\)[/tex].
- Therefore, [tex]\(g(x)\)[/tex] is never zero.
Since [tex]\(g(x)\)[/tex] is never zero, there are no restrictions on [tex]\(x\)[/tex] from [tex]\(g(x)\)[/tex]. Thus, there is no need to exclude any values of [tex]\(x\)[/tex] to keep the denominator non-zero.
3. Conclude the domain:
- Given that [tex]\(g(x) \neq 0\)[/tex] for all real numbers, [tex]\(\frac{f(x)}{g(x)}\)[/tex] or [tex]\(\frac{x}{1} = x\)[/tex] is defined for all real numbers.
Hence, the domain of [tex]\(\left(\frac{f}{g}\right)(x) \)[/tex] is:
[tex]\[
\text{All real numbers}
\][/tex]
