Chapter 6, Problem 61E

Solution

To find the if it is better to use the coupon before or after the store discount.

It is better to use the coupon after the store discount.

Given information:

A store offers 50% discount off a $4.00 box of cereal, and the person has a $1 coupon off for the same.

Calculation:

Let $x$ be the price of the cereal.

Then, the function representing the price after store discount is $f\left(x\right)=0.5x$ .

If the person applies the coupon on actual price of the cereal, the price of the cereal will be $g\left(x\right)=x-1$ .

Now, the price of cereal on applying the discount coupon before the store discount is given by $\left(f\circ g\right)\left(x\right)$ and the price when the coupon applied after the store discount is given by $\left(g\circ f\right)\left(x\right)$ .

Now, find the functions $\left(f\circ g\right)\left(x\right)$ and $\left(g\circ f\right)\left(x\right)$ .

  $\begin{array}{c}\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)\\ =f\left(x-1\right)\\ =0.5\left(x-1\right)\\ \left(g\circ f\right)\left(x\right)=g\left(f\left(x\right)\right)\\ =g\left(0.5x\right)\\ =0.5x-1\end{array}$

Now, find the values when $x=4$ .

  $\begin{array}{c}\left(f\circ g\right)\left(4\right)=0.5\left(4-1\right)\\ =0.5\cdot 3\\ =1.5\\ \left(g\circ f\right)\left(4\right)=0.5\cdot 4-1\\ =2-1\\ =1\end{array}$

This shows that the price of cereal on applying the discount coupon before the store discount is $1.5 and the price when the coupon applied after the store discount is $1. Therefore, it is better to use the coupon after the store discount.

Conclusion:

It is better to use the coupon after the store discount.