Answer :
To determine the restrictions on the domain of the composition function [tex]\((u \circ v)(x)\)[/tex], we need to understand the constraints on both [tex]\( u(x) \)[/tex] and [tex]\( v(x) \)[/tex] as well as how these constraints interact when composing the functions.
### Step-by-Step Solution:
1. Identify the Domains:
- The domain of [tex]\( u(x) \)[/tex] is the set of all real numbers 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 numbers except 2. This means [tex]\( v(x) \)[/tex] is undefined at [tex]\( x = 2 \)[/tex].
2. Determine How Composition Affects the Domain:
- First, consider the function [tex]\( v(x) \)[/tex]. Because [tex]\( v(x) \)[/tex] is a part of the composite function [tex]\( (u \circ v)(x) = u(v(x)) \)[/tex], the value [tex]\( x = 2 \)[/tex] must be excluded from the domain of [tex]\( (u \circ v)(x) \)[/tex]. This is because [tex]\( v(x) \)[/tex] is undefined at [tex]\( x = 2 \)[/tex].
- Next, consider where [tex]\( u(v(x)) \)[/tex] is defined. For [tex]\( u(v(x)) \)[/tex] to be defined, [tex]\( v(x) \)[/tex] must produce values that are within the domain of [tex]\( u(x) \)[/tex]. Since [tex]\( u(x) \)[/tex] is undefined at 0, we must ensure that [tex]\( v(x) \)[/tex] does not equal 0. Thus, we need [tex]\( v(x) \)[/tex] to be different from 0 for [tex]\( (u \circ v)(x) \)[/tex] to be defined.
3. Synthesize the Restrictions:
- The domain of [tex]\( (u \circ v)(x) \)[/tex] excludes values of [tex]\( x \)[/tex] where [tex]\( v(x) \)[/tex] is undefined. Thus, [tex]\( x \neq 2 \)[/tex].
- Additionally, the values of [tex]\( x \)[/tex] for which [tex]\( v(x) = 0 \)[/tex] also need to be excluded because [tex]\( u(x) \)[/tex] is undefined at 0. So, we need to find which [tex]\( x \)[/tex] values make [tex]\( v(x) = 0 \)[/tex].
### Conclusion:
In summary, the restrictions on the domain of [tex]\( (u \circ v)(x) \)[/tex] are:
- [tex]\( x \neq 2 \)[/tex] since [tex]\( v(x) \)[/tex] is undefined at [tex]\( x = 2 \)[/tex].
- [tex]\( x \)[/tex] cannot be any value for which [tex]\( v(x) = 0 \)[/tex] because [tex]\( u(0) \)[/tex] is undefined.
Thus, based on these considerations, the correct restrictions to place on the domain of [tex]\( (u \circ v)(x) \)[/tex] are:
[tex]\[ x \neq 2 \][/tex]
and
\[ x \) cannot be any value for which [tex]\( v(x) = 0 \)[/tex].
This leads us to the conclusion that the correct choice is:
\[ x \neq 2 \) and [tex]\( x \)[/tex] cannot be any value for which [tex]\( v(x) = 0 \)[/tex].
### Step-by-Step Solution:
1. Identify the Domains:
- The domain of [tex]\( u(x) \)[/tex] is the set of all real numbers 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 numbers except 2. This means [tex]\( v(x) \)[/tex] is undefined at [tex]\( x = 2 \)[/tex].
2. Determine How Composition Affects the Domain:
- First, consider the function [tex]\( v(x) \)[/tex]. Because [tex]\( v(x) \)[/tex] is a part of the composite function [tex]\( (u \circ v)(x) = u(v(x)) \)[/tex], the value [tex]\( x = 2 \)[/tex] must be excluded from the domain of [tex]\( (u \circ v)(x) \)[/tex]. This is because [tex]\( v(x) \)[/tex] is undefined at [tex]\( x = 2 \)[/tex].
- Next, consider where [tex]\( u(v(x)) \)[/tex] is defined. For [tex]\( u(v(x)) \)[/tex] to be defined, [tex]\( v(x) \)[/tex] must produce values that are within the domain of [tex]\( u(x) \)[/tex]. Since [tex]\( u(x) \)[/tex] is undefined at 0, we must ensure that [tex]\( v(x) \)[/tex] does not equal 0. Thus, we need [tex]\( v(x) \)[/tex] to be different from 0 for [tex]\( (u \circ v)(x) \)[/tex] to be defined.
3. Synthesize the Restrictions:
- The domain of [tex]\( (u \circ v)(x) \)[/tex] excludes values of [tex]\( x \)[/tex] where [tex]\( v(x) \)[/tex] is undefined. Thus, [tex]\( x \neq 2 \)[/tex].
- Additionally, the values of [tex]\( x \)[/tex] for which [tex]\( v(x) = 0 \)[/tex] also need to be excluded because [tex]\( u(x) \)[/tex] is undefined at 0. So, we need to find which [tex]\( x \)[/tex] values make [tex]\( v(x) = 0 \)[/tex].
### Conclusion:
In summary, the restrictions on the domain of [tex]\( (u \circ v)(x) \)[/tex] are:
- [tex]\( x \neq 2 \)[/tex] since [tex]\( v(x) \)[/tex] is undefined at [tex]\( x = 2 \)[/tex].
- [tex]\( x \)[/tex] cannot be any value for which [tex]\( v(x) = 0 \)[/tex] because [tex]\( u(0) \)[/tex] is undefined.
Thus, based on these considerations, the correct restrictions to place on the domain of [tex]\( (u \circ v)(x) \)[/tex] are:
[tex]\[ x \neq 2 \][/tex]
and
\[ x \) cannot be any value for which [tex]\( v(x) = 0 \)[/tex].
This leads us to the conclusion that the correct choice is:
\[ x \neq 2 \) and [tex]\( x \)[/tex] cannot be any value for which [tex]\( v(x) = 0 \)[/tex].
