2) A car dealership advertises a 15% discount on all its new cars. In addition, the manufacturer offers a $1000 rebate on the purchase of a new car. Let x represent the sticker price of the car.
a. Suppose only the 15% discount applies. Find a function f that models the purchase price of the car as a function of the sticker price x.
b. Suppose only the $1000 rebate applies. Find a function g that models the purchase price of the car as a function of the sticker price x.
c. Find a formula for H=fᵒ g.
d. Find H-1. What does H-1 represent?
e. Find H-1(13,000). What does your answer represent?
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a. To find a function that models the purchase price of the car with only the 15% discount, we need to calculate the amount of the discount and subtract it from the sticker price. The discount is 15% of the sticker price, so the purchase price, P, can be calculated as:
P = x - 0.15x
= 0.85x
Therefore, the function f(x) = 0.85x models the purchase price of the car with only the 15% discount.
b. To find a function that models the purchase price of the car with only the $1000 rebate, we just need to subtract the rebate from the sticker price. The purchase price, P, can be expressed as:
P = x - $1000
Therefore, the function g(x) = x - $1000 models the purchase price of the car with only the $1000 rebate.
c. To find a formula for H = f∘g, we need to substitute g(x) into f(x). Since g(x) = x - $1000, we can substitute this expression into f(x) to get:
H(x) = f(g(x))
= f(x - $1000)
= 0.85(x - $1000)
= 0.85x - $850
Therefore, the formula for H is H(x) = 0.85x - $850.
d. H-1 represents the inverse of the function H. In other words, given a purchase price, H-1 allows us to determine the corresponding sticker price of the car.
e. To find H-1(13,000), we need to substitute the purchase price of $13,000 into the inverse function, H-1, and solve for the corresponding sticker price:
13,000 = 0.85x - $850
To isolate x, we add $850 to both sides:
13,000 + $850 = 0.85x
13,850 = 0.85x
To get x alone, divide both sides by 0.85:
x = 13,850 / 0.85
Using a calculator, we find that x is approximately $16,294.12.
Therefore, H-1(13,000) represents the sticker price of the car that corresponds to a purchase price of $13,000.