To find the domain of the function [tex]\((u \bullet v)(x)\)[/tex], we need to determine the values of [tex]\(x\)[/tex] that are allowed in both [tex]\(u(x)\)[/tex] and [tex]\(v(x)\)[/tex]. This means identifying and excluding any values that would make either function undefined.
Given that:
- The domain of [tex]\(u(x)\)[/tex] is all real numbers except [tex]\(0\)[/tex]. That means [tex]\(x \neq 0\)[/tex] for [tex]\(u(x)\)[/tex] to be defined.
- The domain of [tex]\(v(x)\)[/tex] is all real numbers except [tex]\(2\)[/tex]. That means [tex]\(x \neq 2\)[/tex] for [tex]\(v(x)\)[/tex] to be defined.
To determine the domain of [tex]\((u \bullet v)(x)\)[/tex], we need to take the intersection of these domains. This means we must exclude any [tex]\(x\)[/tex] that makes either function undefined.
So, [tex]\(x \neq 0\)[/tex] (from the restriction on [tex]\(u(x)\)[/tex]) and [tex]\(x \neq 2\)[/tex] (from the restriction on [tex]\(v(x)\)[/tex]).
Hence, the restrictions on the domain of [tex]\((u \bullet v)(x)\)[/tex] are:
- [tex]\(x \neq 0\)[/tex]
- [tex]\(x \neq 2\)[/tex]
Therefore, the correct restrictions on the domain are: [tex]\( x \neq 0 \text{ and } x \neq 2\)[/tex].
