Answer
$(a)$ $f(x)=0.85x$
$(b)$ $g(x)=x-1000$
$(c)$ $H=0.85x-850$
$(d)$ $H^{-1}=\frac{20}{17}x+1000$
$(e)$ $\approx\$16284.12$
It's the original price of the car for given purchase price $\$13000$
Work Step by Step
$(a)$ If $x$ is the price of a car, then the amount actually paid for the car will be $85\%$ of actual price. That is represented by the following function:
$f(x)=0.85x$
$(b)$ If only $\$1000$ rebate is applied, then we simply subtract the rebate from the price of the car:
$g(x)=x-1000$
$(c)$
$f◦g=f(g(x)) = 0.85(x-1000)=0.85x-850$
$(d)$
For $H^{-1}$, we first write it in terms of $x$ and $y$ and then replace $x$ by $y$ and vice versa. Then find $y$ in terms of $x$:
$y=0.85x-850$
$x=0.85y-850$
$0.85y=x+850$
$y=\frac{x+850}{0.85}$
$y=\frac{100x+85000}{85}$
$y=\frac{100x}{85}+\frac{85000}{85}$
$y=\frac{20}{17}x+1000$
$H^{-1}=\frac{20}{17}x+1000$
$(e)$
$H^{-1}(13000)=\frac{20}{17}\times13000+1000\approx16284.12$
It's the original price of the car for given purchase price $\$13000$