Get the most out of your questions with the extensive resources available on IDNLearn.com. Our community provides timely and precise responses to help you understand and solve any issue you face.

The domain of [tex]u(x)[/tex] is the set of all real values except 0, and the domain of [tex]v(x)[/tex] is the set of all real values except 2. What are the restrictions on the domain of [tex](u \cdot v)(x)[/tex]?

A. [tex]u(x) \neq 0[/tex] and [tex]v(x) \neq 2[/tex]

B. [tex]x \neq 0[/tex] and [tex]x[/tex] cannot be any value for which [tex]u(x) = 2[/tex]

C. [tex]x \neq 2[/tex] and [tex]x[/tex] cannot be any value for which [tex]v(x) = 0[/tex]

D. [tex]u(x) \approx 2[/tex] and [tex]v(x) \approx 0[/tex]


Sagot :

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].