To determine the restrictions of the domain of the composite function [tex]\( f(g(x)) \)[/tex], we need to follow these steps:
1. Understand the individual functions:
- The function [tex]\( f(x) = \frac{1}{x+5} \)[/tex] has a restriction that its denominator must not be zero. Therefore, [tex]\( x + 5 \neq 0 \)[/tex], which implies [tex]\( x \neq -5 \)[/tex].
- The function [tex]\( g(x) = x - 2 \)[/tex] does not have any restrictions on its domain since it is a simple linear function.
2. Form the composite function [tex]\( f(g(x)) \)[/tex]:
- Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[
f(g(x)) = f(x-2) = \frac{1}{(x-2) + 5}
\][/tex]
- Simplify the expression:
[tex]\[
f(g(x)) = \frac{1}{x-2+5} = \frac{1}{x+3}
\][/tex]
3. Determine the restriction for the composite function:
- The function [tex]\( \frac{1}{x+3} \)[/tex] has a restriction that the denominator must not be zero. Therefore:
[tex]\[
x + 3 \neq 0
\][/tex]
- Solving for [tex]\( x \)[/tex], we get:
[tex]\[
x \neq -3
\][/tex]
Therefore, the restriction for the domain of the composite function [tex]\( f(g(x)) \)[/tex] is [tex]\( x \neq -3 \)[/tex].
So, the correct answer is:
[tex]\[ x \neq -3 \][/tex]
