Q. 75 - Precalculus Enhanced with Graphing Utilities

Question

Discounts The manufacturer of a computer is offering two discounts on last year’s model computer. The first discount is a $200 rebate and the second discount is 20% off the regular price, p.

(a) Write a function f that represents the sale price if only the rebate applies.

(b) Write a function g that represents the sale price if only the 20% discount applies.

(c) Find $f\circ g$ and $g\circ f$ . What does each of these functions represent? Which combination of discounts represents a better deal for the consumer? Why?

Step-by-Step Solution

Answer

(a) $f\left(p\right)=p-200$

(b)$g\left(p\right)=0.8p$

(c) $f\circ g=0.8p-200$ and $g\circ f=0.8p-160$

Step 1. Given Information

The manufacturer of a computer is offering two discounts on last year’s model computer. The first discount is a $200 rebate and the second discount is 20% off the regular price, p.

Part (a) Step 1. Finding a function f that represents the sale price if only the rebate applies.

Let $p=$regular price of the computer

The rebate offered is$\$200.$ .Thus, the function that will represent the sale price if the rebate$\$200$ is applied is one where is subtracted from the regular price of the computer, that

is:

$f\left(p\right)=p-200$

Part (b) Step 1. A function g that represents the sale price if 20% discount applies

The function that represents the sale price if only the 20%discount is one where 20% is taken off the regular price of the computer.

$$\begin{gathered} g\left(p\right)=\left(100-20\right)\%p \\ g\left(p\right)=80\%p \end{gathered}$$

Part (c) Step 1. Finding f ∘ g

$$\begin{gathered} f\circ g=f\left(g\left(p\right)\right)=f\left(0.8p\right) \\ f\circ g=0.8p-200 \end{gathered}$$

Part (c) Step 2. Finding g ∘ f

$$\begin{gathered} g\circ f=g\left(f\left(x\right)\right)=g\left(f\left(p\right)\right) \\ g\circ f=g\left(p-200\right) \\ g\circ f=0.8\left(p-200\right) \\ g\circ f=0.8p-160 \end{gathered}$$