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Answer :
To solve this problem, let's start by understanding the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].
### Part a: Describing the Functions
1. Function [tex]\( f(x) = x - 170 \)[/tex]:
This function models the price of the computer after a [tex]$170 discount. When you subtract 170 from the original price \( x \), you're lowering the price by that fixed amount.
2. Function \( g(x) = 0.75x \):
This function models the price of the computer after a 25% discount. When you multiply the original price \( x \) by 0.75, you're effectively paying 75% of the original price, which is the same as getting a 25% discount.
### Part b: Finding and Describing \( (f \circ g)(x) \)
To find \( (f \circ g)(x) \), we need to calculate \( f(g(x)) \).
1. Apply \( g(x) \) first:
We start with \( g(x) = 0.75x \).
2. Then apply \( f(x) \) to \( g(x) \):
\( f(g(x)) = f(0.75x) \).
Substitute \( g(x) = 0.75x \) into \( f(x) = x - 170 \):
\[
f(g(x)) = 0.75x - 170
\]
The expression \( 0.75x - 170 \) represents the price of the computer after both a 25% discount and a $[/tex]170 discount have been applied successively. This compounded price reduction first reduces the original price by 25%, and then subtracts an additional $170 from that reduced amount.
### Part a: Describing the Functions
1. Function [tex]\( f(x) = x - 170 \)[/tex]:
This function models the price of the computer after a [tex]$170 discount. When you subtract 170 from the original price \( x \), you're lowering the price by that fixed amount.
2. Function \( g(x) = 0.75x \):
This function models the price of the computer after a 25% discount. When you multiply the original price \( x \) by 0.75, you're effectively paying 75% of the original price, which is the same as getting a 25% discount.
### Part b: Finding and Describing \( (f \circ g)(x) \)
To find \( (f \circ g)(x) \), we need to calculate \( f(g(x)) \).
1. Apply \( g(x) \) first:
We start with \( g(x) = 0.75x \).
2. Then apply \( f(x) \) to \( g(x) \):
\( f(g(x)) = f(0.75x) \).
Substitute \( g(x) = 0.75x \) into \( f(x) = x - 170 \):
\[
f(g(x)) = 0.75x - 170
\]
The expression \( 0.75x - 170 \) represents the price of the computer after both a 25% discount and a $[/tex]170 discount have been applied successively. This compounded price reduction first reduces the original price by 25%, and then subtracts an additional $170 from that reduced amount.
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