) 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?
Fg= 12
a.30
a. The purchase price of the car after a 15% discount can be calculated by subtracting 15% of the sticker price from the sticker price itself. Let's represent the purchase price as P.
P = x - 0.15x
Simplifying the equation, we get:
P = 0.85x
Therefore, the function f that models the purchase price of the car as a function of the sticker price x is:
f(x) = 0.85x
b. The purchase price of the car after a $1000 rebate can be calculated by subtracting $1000 from the sticker price. Let's represent the purchase price as Q.
Q = x - 1000
Therefore, the function g that models the purchase price of the car as a function of the sticker price x is:
g(x) = x - 1000
c. The composite function H = fᵒg can be obtained by substituting the function g(x) into the function f(x) as follows:
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 function of H. It represents a function that can take the purchase price of a car and determine the sticker price.
e. To find H⁻¹(13,000), we substitute 13,000 into the formula for H⁻¹:
H⁻¹(13,000) = (13,000 + 850) / 0.85
Simplifying the equation, we get:
H⁻¹(13,000) = 15,411.76
Therefore, H⁻¹(13,000) represents the sticker price of a car when the purchase price is $13,000.
a. To find a function that models the purchase price of the car as a function of the sticker price x when only the 15% discount applies, we need to subtract the discount amount from the sticker price.
The 15% discount can be expressed as 0.15 times the sticker price (0.15 * x). So, the function f(x) = x - 0.15x = 0.85x.
b. To find a function that models the purchase price of the car as a function of the sticker price x when only the $1000 rebate applies, we need to subtract the rebate amount from the sticker price.
The function g(x) = x - $1000.
c. The function H(fᵒ g) represents the composition of functions f and g. To find H, we substitute the function g(x) into the function f(x).
H(x) = f(g(x)) = f(x - $1000) = 0.85(x - $1000)
d. To find H-1, we need to find the inverse function of H. This means finding a function that undoes the operations performed by H.
To find the inverse, we can swap the roles of x and H(x) and solve for x.
H-1(x) = (x + $1000) / 0.85
e. To find H-1(13,000), we substitute 13,000 into the inverse function H-1(x):
H-1(13,000) = (13,000 + $1000) / 0.85 = $16,000
So, H-1(13,000) represents the sticker price of the car before any discounts or rebates were applied.