Answer
a. $f(x)=0.85x$
b. $g(x)=x-1000$.
c. $H=0.85x-850$
d. $H^{-1}(x)=1.176x+1000$
e. $16,288$
Work Step by Step
a.
A function that models the purchase price of a car if only the $15\%$ discount applies as a function of the sticker price x is, $f(x)=0.85x$.
b.
A function that models the purchase price of a car if only the $1000$ rebate applies as a function of the sticker price x is, $g(x)=x-1000$.
c.
$H=f \circ g$,
$H=0.85(x-1000)=0.85x-850$
d.
$0.85x-850=y$,
$0.85x=y+850$,
$x=\frac{y+850}{0.85}$,
$\frac{x}{0.85}+1000=H^{-1}(x)$,
$H^{-1}(x)=1.176x+1000$
$H^{-1}$ models the sticker price of the car as a function of a purchase price when both $15\%$ discount and $1000$ rebate apply.
e.
$H^{-1}(13,000)=1.176(13,000)+1000=16,288$
This is the original price of the car when a rebate of $1000$ and a discount of $15\%$ (which is $13,000$) apply.
