To determine the domain of the function [tex]\((f \times g)(x)\)[/tex], where [tex]\(f(x) = x + 7\)[/tex] and [tex]\(g(x) = \frac{1}{x - 13}\)[/tex], we need to consider the domains of the individual functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] and find their intersection within the context of the combined function [tex]\((f \times g)(x)\)[/tex].
1. Domain of [tex]\(f(x) = x + 7\)[/tex]:
- The function [tex]\(f(x) = x + 7\)[/tex] is a linear function.
- Linear functions are defined for all real numbers.
- Hence, the domain of [tex]\(f(x)\)[/tex] is all real numbers, i.e., [tex]\(\mathbb{R}\)[/tex].
2. Domain of [tex]\(g(x) = \frac{1}{x - 13}\)[/tex]:
- The function [tex]\(g(x) = \frac{1}{x - 13}\)[/tex] is a rational function.
- Rational functions are defined for all real numbers except where the denominator is zero.
- The denominator of [tex]\(g(x)\)[/tex] is [tex]\(x - 13\)[/tex], which is zero when [tex]\(x = 13\)[/tex].
- Hence, the domain of [tex]\(g(x)\)[/tex] is all real numbers except [tex]\(x = 13\)[/tex], i.e., [tex]\(\mathbb{R} \setminus \{13\}\)[/tex].
3. Domain of [tex]\((f \times g)(x) = f(x) \times g(x)\)[/tex]:
- To form [tex]\((f \times g)(x)\)[/tex], we multiply [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]: [tex]\((f \times g)(x) = (x + 7) \times \frac{1}{x - 13}\)[/tex].
- Since [tex]\(f(x)\)[/tex] is defined for all real numbers and [tex]\(g(x)\)[/tex] is defined for all real numbers except [tex]\(x = 13\)[/tex], the product [tex]\((f \times g)(x)\)[/tex] will be defined wherever both functions are defined.
- Therefore, [tex]\((f \times g)(x)\)[/tex] is defined for all real numbers except [tex]\(x = 13\)[/tex].
So, the domain of [tex]\((f \times g)(x)\)[/tex] is:
[tex]\[ x \ne 13 \][/tex]
Thus, the answer is:
[tex]\[ \boxed{x \ne 13} \][/tex]
