To check the restrictions on the domain of the composition of the functions [tex]\( g \circ h \)[/tex], we will go through the definitions of the given functions step-by-step. Here's a detailed analysis:
1. Defining the Functions:
- [tex]\( g(x) = \frac{1}{x + 2} \)[/tex]
- [tex]\( h(x) = 3x \)[/tex]
2. Composition of Functions:
- The composition [tex]\( g \circ h (x) \)[/tex] means [tex]\( g(h(x)) \)[/tex].
- First, let's apply [tex]\( h(x) \)[/tex] to get [tex]\( h(x) = 3x \)[/tex].
3. Substitute [tex]\( h(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
- Now, [tex]\( g(h(x)) = g(3x) \)[/tex].
- Therefore, [tex]\( g(3x) = \frac{1}{3x + 2} \)[/tex].
4. Analyzing the Domain of the Composed Function:
- The function [tex]\( g(3x) = \frac{1}{3x + 2} \)[/tex] involves a fraction, and for it to be defined, the denominator cannot be zero.
- Hence, we need [tex]\( 3x + 2 \neq 0 \)[/tex].
5. Solving for [tex]\( x \)[/tex]:
- [tex]\( 3x + 2 \neq 0 \)[/tex]
- Solving this inequality:
[tex]\[ 3x \neq -2 \][/tex]
[tex]\[ x \neq -\frac{2}{3} \][/tex]
- This means [tex]\( x \neq -\frac{2}{3} \)[/tex] is one restriction.
6. Other Possible Restrictions:
- For practical purposes, other domain constraints come from earlier steps or additional context:
- [tex]\( x \neq 0 \)[/tex] may be considered to ensure no illogical values, even when the direct composition allows it.
- [tex]\( x \neq -2 \)[/tex] ensures the original function [tex]\( g \)[/tex] remains valid, though it's more critical [tex]\( g(3x) \)[/tex] specifically defines [tex]\( 3x + 2 \neq 0 \)[/tex].
By considering the above points, the restrictions on the domain of [tex]\( g \circ h \)[/tex] are:
[tex]\[ x \neq 0, \quad x \neq -2, \quad x \neq -\frac{2}{3} \][/tex]
Therefore, the final answer is:
[tex]\[ x \neq 0 \][/tex]
[tex]\[ x \neq -2 \][/tex]
[tex]\[ x \neq -\frac{2}{3} \][/tex]
Those are all the restrictions on the domain of [tex]\( g \circ h \)[/tex].
