You have a $50 coupon from the

manufacturer good for the purchase of a cell phone. The store
where you are purchasing your cell phone is offering a 20%
discount on all cell phones. Let x represent the regular price of
the cell phone.

(a) Suppose only the 20% discount applies. Find a function f
that models the purchase price of the cell phone

(b) Suppose only the $50 coupon applies. Find a function g
that models the purchase price of the cell phone as a function
of the sticker price x.
(c) If you can use the coupon and the discount, then the purchase
price is either (f o g) (x) or (g o f) (x) , depending on the
order in which they are applied to the price. Find both... Which composition gives the lower

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What would professor Price at FSU do?

(a) If only the 20% discount applies, the purchase price of the cell phone can be calculated as follows:

Discount = 20% of x = 0.2x
Purchase price = x - Discount = x - 0.2x = 0.8x

Therefore, the function f that models the purchase price of the cell phone with only the 20% discount is f(x) = 0.8x.

(b) If only the $50 coupon applies, the purchase price of the cell phone can be calculated as follows:

Purchase price = x - $50

Therefore, the function g that models the purchase price of the cell phone with only the $50 coupon is g(x) = x - $50.

(c) To find the composition (f o g)(x), we need to apply the functions in the order f(g(x)). Substituting g(x) into f(x), we get:

(f o g)(x) = f(g(x)) = f(x - $50) = 0.8(x - $50) = 0.8x - $40.

To find the composition (g o f)(x), we need to apply the functions in the order g(f(x)). Substituting f(x) into g(x), we get:

(g o f)(x) = g(f(x)) = g(0.8x) = 0.8x - $50.

To determine which composition gives the lower price, we can compare the two functions:

0.8x - $40 vs. 0.8x - $50

Since the $10 discount from the coupon is greater than the $10 difference in the compositions, using the coupon before the discount (g o f)(x) results in a lower price. Therefore, the composition (g o f)(x) gives the lower purchase price.

(a) If only the 20% discount applies, the purchase price of the cell phone can be modeled by the function f(x) = x - 0.20x = 0.80x.

Explanation: To calculate the purchase price with a 20% discount, you need to subtract 20% of the regular price (0.20x) from the regular price (x). This gives you a purchase price of 80% (or 0.80) of the regular price.

(b) If only the $50 coupon applies, the purchase price of the cell phone can be modeled by the function g(x) = x - $50.

Explanation: To calculate the purchase price with the $50 coupon, you simply subtract $50 from the sticker price (x).

(c) If you can use both the coupon and the discount, then the purchase price can be calculated using the composition of functions (f o g)(x) or (g o f)(x).

(f o g)(x) = f(g(x)) = f(x - $50) = (x - $50) - 0.20(x - $50) = x - $50 - 0.20x + $10 = 0.80x - $40.

(g o f)(x) = g(f(x)) = g(0.80x) = 0.80x - $50.

Explanation: In the composition (f o g)(x), we first apply the $50 coupon to the sticker price (x), which gives us x - $50. Then, we apply the 20% discount to the resulting price, which gives us 0.80(x - $50). Simplifying this expression, we get 0.80x - $40.

In the composition (g o f)(x), we first apply the 20% discount to the regular price (x), which gives us 0.80x. Then, we apply the $50 coupon to the resulting price, which gives us 0.80x - $50.

To determine which composition gives the lower purchase price, we compare 0.80x - $40 with 0.80x - $50. Since $40 is less than $50, the composition (f o g)(x) gives the lower purchase price.