The prompt: Amparo bought a jacket with a \$50 gift certificate she received as a birthday present. The jacket was marked 33% off, and the sales tax in her area is 5.5%. If she paid $45.95 for the jacket, use composition of functions to determine the original price of the jacket. (45.95 was the price after they redeemed the gift certificate)
a.) Write a function $g(x)$ for the price after the gift certificate was used and then determine the price before the gift certificate was used.
b.) Write s function $t(x)$ for the price after the discount was applied and then determine the original price/
c.) Write a function $d(x)$ for the price after the discount was applied and then determine the original price.
d.) Write a simplified composition of functions $p(x)=g(t(d(x)))$ to find the price of any vase where the discount, sales tax, and gift certificate are all accounted for.
e.) If Amparo wanted to get a vase at no additional cost after all discounts, taxes, and their gift certificate was redeemed, what would be the original price of the most expensive vase they could buy?
I am not sure where to start exactly with this question since it needs to be written as a function. I am not sure if this is right, but would you just add 50 to problem a?
a.) $g(x)=45.95x-50$
I am pretty sure that is incorrect, but I am extremely confused. If anyone could help me solve the problems that would be amazing. Thank you so much.
$g(x) = x-50\\ g^{-1}(x) = 50 + x$
The gift certificate takes $\$50$ of the final price. $g^{-1}(x)$ is the inverse function that tells us the price before the certificate is applied
$t(x)$ is probably taxes, and not the discount applied twice
$t(x) = 1.055 x$
$d(x) = (1-0.33)x = 0.67 x$
$p(x) = (g\circ t\circ d) (x) = (1.055)(0.67) x - 50\\ p^{-1}(x) = (d^{-1}\circ t^{-1}\circ g^{-1})(x) = \frac {x+50}{(1.055)(0.67)}$