Answer to Question #284824 in Calculus for Jarah

Question

Suppose you are buying face mask for yourself, your friends, and family during this covid-19

pandemic. The face mask shop has a deal going, if you buy one facemask for 35 pesos, then

additional face masks are only 30 pesos each. Using a short bond paper, do the following:

a. Represent this situation into a rational equation showing the price per face mask based

on the number of face masks.

b. Determine the horizontal asymptote and explain what the horizontal asymptote

represents.

c. Graph the function appropriately and determine its domain and range.

d. Is the original function one-to-one? Explain

Answer 1

a. Let $x=$ the number of face masks, $x\in \N$. Let $p(x)=$ the price per face mask in pesos. Then

$$p(x)=\dfrac{35+30(x-1)}{x}$$

$$=\dfrac{30x+5}{x}$$

$$=30+\dfrac{5}{x},\ x\in \N$$

b.

$p(x)\to30$ as $x\to\infin.$

Horizontal asymptote $p(x)=30.$

When we buy infinitely much face masks, the face masks are approximately 30 pesos each.

c.

Domain: $x\in\N$

Range: $(30, 35]$

d.

Use the Horizontal Line Test.

No horizontal line intersects the graph of the function $p(x)$ in more than one point, then the function is one-to-one.

$$p(x_1)=p(x_2)=>30+\dfrac{5}{x_1}=30+\dfrac{5}{x_2}$$

$$=>\dfrac{5}{x_1}=\dfrac{5}{x_2}=>\dfrac{1}{x_1}=\dfrac{1}{x_2}$$

$$=>x_1=x_2, (x_1, x_2\in \N)$$

Therefore, the function $p(x)$ is one-to-one.