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3. Julie works at an electronics store. She receives a weekly salary and a [tex]5\%[/tex] commission on her sales over [tex]\$4,000[/tex]. The function for her commission is [tex]f(x) = 0.05x[/tex], where [tex]x[/tex] is her total sales. The function representing how much of her sales she earns commission on is [tex]g(x) = x - 4000[/tex]. If her commission is determined by finding [tex]f(g(x))[/tex], what is the function to determine her commission?

A. [tex]f(g(x)) = 0.05x + 200[/tex]
B. [tex]J(g(x)) = 0.05x - 4000[/tex]
C. [tex]f(g(z)) = 0.05x + 4000[/tex]
D. [tex]f(g(x)) = 0.05x - 200[/tex]



Answer :

GinnyAnswer
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]

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