To determine the restrictions on the domain of the product function [tex]\((u \cdot v)(x)\)[/tex], we need to consider the domains of the individual functions [tex]\(u(x)\)[/tex] and [tex]\(v(x)\)[/tex].
1. Domains of Individual Functions:
- The domain of [tex]\(u(x)\)[/tex] is the set of all real values except 0. This means [tex]\(u(x)\)[/tex] is undefined at [tex]\(x = 0\)[/tex].
- The domain of [tex]\(v(x)\)[/tex] is the set of all real values except 2. This means [tex]\(v(x)\)[/tex] is undefined at [tex]\(x = 2\)[/tex].
2. Finding the Intersection:
- The domain of [tex]\((u \cdot v)(x)\)[/tex] will be the intersection of the domains of [tex]\(u(x)\)[/tex] and [tex]\(v(x)\)[/tex]. This means that [tex]\((u \cdot v)(x)\)[/tex] will be defined only where both [tex]\(u(x)\)[/tex] and [tex]\(v(x)\)[/tex] are defined.
3. Restrictions Analysis:
- Since [tex]\(u(x)\)[/tex] is undefined at [tex]\(x = 0\)[/tex], [tex]\((u \cdot v)(x)\)[/tex] will also be undefined at [tex]\(x = 0\)[/tex].
- Since [tex]\(v(x)\)[/tex] is undefined at [tex]\(x = 2\)[/tex], [tex]\((u \cdot v)(x)\)[/tex] will also be undefined at [tex]\(x = 2\)[/tex].
Therefore, the domain of [tex]\((u \cdot v)(x)\)[/tex] is all real values except [tex]\(x = 0\)[/tex] and [tex]\(x = 2\)[/tex].
Conclusion:
The restrictions on the domain of [tex]\((u \cdot v)(x)\)[/tex] are:
- [tex]\(x \neq 0\)[/tex] (because [tex]\(u(x)\)[/tex] is undefined at 0)
- [tex]\(x \neq 2\)[/tex] (because [tex]\(v(x)\)[/tex] is undefined at 2)
The correct answer is:
[tex]\[ \boxed{x \neq 0 \text{ and } x \neq 2} \][/tex]
Hence, the domain of [tex]\((u \cdot v)(x)\)[/tex] is all real values except [tex]\(x = 0\)[/tex] and [tex]\(x = 2\)[/tex].
