Question

In the mail, you receive a coupon for $5 off of a pair of jeans. When you arrive at the store, you find that all jeans are 25% off. Define two functions that represent the two types of discounts separately. Let f(x) be the dollars off and g(x) be the percent off. Then choose which composition of functions gives you the best price for a $40 pair of jeans. Then state what that best price is exactly.

Answer 1

To define the functions for the discounts, we can represent the two types of discounts as follows:

1. Function f(x) - This function represents the dollars off given by the coupon.

   • Let $f(x)=5$ (this means you get $5 off regardless of the initial price).

2. Function g(x) - This function represents the percentage off the jeans.

   • For a pair of jeans that costs $x$ dollars, the function can be defined as: $$g(x)=0.25\times x$$ This means that you get 25% off the price $x$.

Now, let's find the final price of the jeans using both types of discounts for a pair of jeans that costs $40.

Option 1: Using the dollar off coupon first (f(x) first)

1. Apply f(x):

   • Start with the full price: $40$
   • Apply the coupon: $$P_1=40-f(40)=40-5=35$$

2. Then apply g(x):

   • Apply the 25% discount on the new price: $$g(35)=0.25\times 35=8.75$$
   • The final price after both discounts: $$P_1^{\prime }=35-g(35)=35-8.75=26.25$$

Option 2: Applying the percentage discount first (g(x) first)

1. Apply g(x):

   • Calculate 25% off the original price: $$g(40)=0.25\times 40=10$$
   • The new price after the 25% discount: $$P_2=40-g(40)=40-10=30$$

2. Then apply f(x):

   • Now apply the $5 off coupon on the reduced price: $$P_2^{\prime }=30-f(30)=30-5=25$$

Conclusion

Now we compare the final prices from both options:

• Option 1 gave a final price of $26.25.
• Option 2 gave a final price of $25.00.

The best price for the pair of jeans is thus $25.00, obtained by applying the 25% discount first followed by the $5 off coupon.