Answer to Question #239086 in Calculus for Elsa

Question

Question 2

When two given functions, 𝑓(𝑥) = 𝑥2 + 3 𝑎𝑛𝑑 𝑔(𝑥) = 2𝑥 − 5, are divided with each other, a new third function is obtained. Find (𝑓/𝑔)(𝑥) and identify its domain.

Question 3

A new pizzeria has opened near you home, the owner has decided to offer a 20% discount on the opening day on the pizza whose original price is x dollars. The owner is offering additional 15% discount coupon to those who agree to fill a feedback form afterwards. You and your friends have agreed to provide feedback and thus have become entitled to the additional 15% discount coupon. Use the concept of composite functions to identify the amount that you will pay after applying the coupon on the sale price.

Answer 1

Question 2

$(𝑓/𝑔)(𝑥) = \dfrac{ 𝑥^2 + 3 }{ 2𝑥 − 5}$

The numerator is defined on the whole real numbers, but the denominator should not be equal to zero. We should solve an equation $2x-5=0$ to find points where the denominator is 0.

$2x-5=0 \; \Rightarrow \; 2x = 5 \; \Rightarrow \; x = 2.5$.

Therefore, $\mathrm{dom} (f/g)(x) = \mathbb{R} \setminus \lbrace2.5\rbrace$ .

Question 3

Let x be the initial price of pizza. The discount is 20%, so the real price of pizza is $100\%-20\% =80\%$ from the initial price, or 0.80x.

Next, there are two possible options.

1) If the discount coupon is applied to the initial price and is combined with a discount of 20%, we get the total discount 35% and we pay only 0.65x.

2) If the discount coupon is applied to price after applying a 20% discount, we obtain the final price $0.8x - 0.15\cdot 0.8x = 0.85\cdot0.8x = 0.68x$, so the first discount is 20% from the initial price and the second discount is 15% of 0.8x or 12% from the initial price and the total discount is 32%.