To determine Julie's commission, we need to compose the functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] given in the problem.
1. Identify the given functions:
[tex]\[
f(x) = 0.05x
\][/tex]
This represents the commission rate of [tex]\( 5\% \)[/tex] on a certain sales amount.
[tex]\[
g(x) = x - 4000
\][/tex]
This function tells us how much of her sales exceed [tex]\( \$4000 \)[/tex].
2. Compose the functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] to find the commission function [tex]\( f(g(x)) \)[/tex]:
[tex]\[
f(g(x)) = f(x - 4000)
\][/tex]
3. Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[
f(g(x)) = f(x - 4000)
\][/tex]
We replace [tex]\( x \)[/tex] in [tex]\( f(x) \)[/tex] with [tex]\( g(x) \)[/tex]:
[tex]\[
f(x - 4000) = 0.05(x - 4000)
\][/tex]
4. Simplify the expression:
[tex]\[
0.05(x - 4000) = 0.05x - 0.05 \times 4000
\][/tex]
[tex]\[
= 0.05x - 200
\][/tex]
So, the function to determine Julie's commission is:
[tex]\[
f(g(x)) = 0.05x - 200
\][/tex]
Thus, the correct choice from the options given is:
[tex]\[
f(g(x)) = 0.05 x - 200
\][/tex]
