Let's analyze how to calculate the total cost after applying both the rebate and the discount on the original price of [tex]$\$[/tex] 38[tex]$. We need to compute two compositions: the rebate after the discount ($[/tex]r(p(x))[tex]$) and the discount after the rebate ($[/tex]p(r(x))[tex]$).
1. Step-by-Step Calculation:
a. Calculating Rebate and Discount Individually:
- Original price: \$[/tex]38
- Rebate: \[tex]$5
- Discount percentage: 25% (or 0.25 in decimal form)
b. Applying Rebate First $[/tex](r(x))[tex]$:
- Price after rebate: \(r(x) = \text{original price} - \text{rebate} = 38 - 5 = 33\)
c. Applying Discount First $[/tex](p(x))[tex]$:
- Price after discount: \(p(x) = \text{original price} - 25\% \text{ of original price} = 38 - (0.25 \times 38) = 38 - 9.5 = 28.5\)
2. Compositions:
a. Rebate after Discount $[/tex](r(p(x)))[tex]$:
- First, apply the discount: 28.5 (as calculated)
- Then, apply the rebate: \(r(p(x)) = \text{price after discount} - \text{rebate} = 28.5 - 5 = 23.5\)
b. Discount after Rebate $[/tex](p(r(x)))[tex]$:
- First, apply the rebate: 33 (as calculated)
- Then, apply the discount: \(p(r(x)) = \text{price after rebate} - 25\% \text{ of price after rebate} = 33 - (0.25 \times 33) = 33 - 8.25 = 24.75\)
3. Comparing Savings:
- Original price: \$[/tex]38
- Savings with [tex]$r(p(x))$[/tex]: [tex]\(\text{savings} = \text{original price} - r(p(x)) = 38 - 23.5 = 14.5\)[/tex]
- Savings with [tex]$p(r(x))$[/tex]: [tex]\(\text{savings} = \text{original price} - p(r(x)) = 38 - 24.75 = 13.25\)[/tex]
From the above calculations, we observe that the composition of rebate after discount [tex]$(r(p(x)))$[/tex] saves \[tex]$14.5, whereas the composition of discount after rebate $[/tex](p(r(x)))[tex]$ saves \$[/tex]13.25.
Therefore, [tex]$r(p(x))$[/tex] will save her the most money, and she will save \$14.50.
