To determine the domain of the function [tex]\((c d)(x)\)[/tex] where [tex]\(c(x) = \frac{5}{x-2}\)[/tex] and [tex]\(d(x) = x + 3\)[/tex], we need to analyze the domains of the individual functions [tex]\(c(x)\)[/tex] and [tex]\(d(x)\)[/tex], and then find the domain of their product.
1. Domain of [tex]\(c(x)\)[/tex]:
- The function [tex]\(c(x) = \frac{5}{x-2}\)[/tex] is a rational function.
- Rational functions are undefined wherever the denominator is zero.
- For [tex]\(c(x)\)[/tex], the denominator [tex]\(x - 2\)[/tex] cannot be zero.
- So, [tex]\(x - 2 \neq 0\)[/tex].
- Therefore, [tex]\(x \neq 2\)[/tex].
- Thus, the domain of [tex]\(c(x)\)[/tex] is all real values of [tex]\(x\)[/tex], except [tex]\(x = 2\)[/tex].
2. Domain of [tex]\(d(x)\)[/tex]:
- The function [tex]\(d(x) = x + 3\)[/tex] is a polynomial function.
- Polynomial functions are defined for all real values of [tex]\(x\)[/tex].
- Therefore, the domain of [tex]\(d(x)\)[/tex] is all real values of [tex]\(x\)[/tex].
3. Domain of [tex]\((c d)(x) = c(x) \cdot d(x)\)[/tex]:
- We need to combine the domains of [tex]\(c(x)\)[/tex] and [tex]\(d(x)\)[/tex] to find any restrictions on [tex]\(x\)[/tex].
- The product [tex]\((c d)(x) = \left(\frac{5}{x-2}\right) \cdot (x+3)\)[/tex] will be undefined wherever [tex]\(c(x)\)[/tex] or [tex]\(d(x)\)[/tex] is undefined.
- From the domain of [tex]\(c(x)\)[/tex], we know [tex]\(x \neq 2\)[/tex].
- Additionally, the function [tex]\((c d)(x)\)[/tex] might introduce new restrictions if the product simplifies to any further rational expressions. In this case, the product itself must be undefined wherever the combined denominator is zero.
4. Analyzing [tex]\((c d)(x)\)[/tex]:
- [tex]\((c d)(x) = \left(\frac{5}{x-2}\right) \cdot (x+3)\)[/tex]
- The product will be undefined if the numerator or denominator terms induce undefined operations.
- Since [tex]\(x = 2\)[/tex] already makes the denominator of [tex]\(c(x)\)[/tex] zero, include [tex]\(x = 2\)[/tex].
- Additionally, [tex]\(x = -3\)[/tex] makes the entire expression potentially zero, depending on the specific question context, which does not seem to apply restrictions beyond [tex]\(x = 2\)[/tex].
Thus, the combined domain restrictions are:
[tex]\[ x \neq 2 \][/tex]
[tex]\[ x \neq -3 \][/tex]
Therefore, the domain of the function [tex]\(((c d)(x))\)[/tex] is all real values of [tex]\(x\)[/tex] except [tex]\(x = 2\)[/tex] and [tex]\(x = -3\)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{\text{all real values of } x \text{ except } x = 2 \text{ and } x = -3} \][/tex]
