Given: [tex]\( f(x) = \frac{1}{x+5} \)[/tex] and [tex]\( g(x) = x-2 \)[/tex]

What are the restrictions on the domain of [tex]\( f(g(x)) \)[/tex]?

A. [tex]\( x \neq -5 \)[/tex]
B. [tex]\( x \neq -3 \)[/tex]
C. [tex]\( x \neq 2 \)[/tex]
D. There are no restrictions.



Answer :

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]