Given the functions [tex]\( f(x) = \frac{1}{x - 5} \)[/tex] and [tex]\( g(x) = 3x + 12 \)[/tex], we want to find the composition [tex]\( (f \circ g)(x) \)[/tex].
The composition [tex]\( (f \circ g)(x) \)[/tex] means we first apply the function [tex]\( g(x) \)[/tex] and then apply the function [tex]\( f \)[/tex] to the result of [tex]\( g(x) \)[/tex].
1. Find [tex]\( g(x) \)[/tex]:
[tex]\[
g(x) = 3x + 12
\][/tex]
2. Substitute [tex]\( g(x) \)[/tex] into [tex]\( f \)[/tex]:
[tex]\[
f(g(x)) = f(3x + 12)
\][/tex]
3. Apply [tex]\( f \)[/tex] to [tex]\( g(x) \)[/tex]:
[tex]\[
f(3x + 12) = \frac{1}{(3x + 12) - 5}
\][/tex]
4. Simplify the denominator:
[tex]\[
(3x + 12) - 5 = 3x + 7
\][/tex]
5. Write the composed function:
[tex]\[
(f \circ g)(x) = \frac{1}{3x + 7}
\][/tex]
So, the composition [tex]\( (f \circ g)(x) \)[/tex] is:
[tex]\[
(f \circ g)(x) = \frac{1}{3x + 7}
\][/tex]
This could be confirmed by plugging the given answer for a specific value of [tex]\( x = 2 \)[/tex], but we already completed the steps to determine the final expression.
