oyce is buying a pair of jeans for $39.99. The jeans are on sale at a 20% discount and the sales tax is 7%. Write a function T(x) to represent the total cost of an item with 7% sales tax. Find the composite function (T of S)(x).Use the composition to find the final cost of the jeans.
Jason bought a vase with $50 gift certificate. The vase was marked 33% off, and the sales tax was 5.5%. If he paid $45.95 for the vase, use the composition of functions to determine the original price of the vase.
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To find the composite function (T of S)(x), we need to consider the following steps:
1. Calculate the discounted price after a 20% discount: To find the discounted price, we multiply the original price by the discount rate (in decimal form). In this case, the original price is $39.99, and the discount rate is 20%, or 0.20. So, the discounted price will be: 39.99 * (1 - 0.20) = $31.99.
2. Calculate the total cost including sales tax: To find the total cost, we add the sales tax to the discounted price. The sales tax is 7%.
3. Write the function T(x) to represent the total cost of an item with 7% sales tax:
T(x) = x + (x * 0.07) = x * 1.07
4. Find the composite function (T of S)(x):
(T of S)(x) = T(S(x)) = T(31.99) = 31.99 * 1.07.
5. Calculate the final cost of the jeans:
The final cost of the jeans is (T of S)(31.99) = 31.99 * 1.07 = $34.27.
For the second question, we need to find the original price of the vase.
1. Calculate the sale price after a 33% discount: To find the sale price, we multiply the original price by the discount rate (in decimal form). In this case, the original price is unknown, and the discount rate is 33%, or 0.33. So, the sale price will be: x * (1 - 0.33).
2. Calculate the total cost including sales tax: To find the total cost, we add the sales tax to the sale price. The sales tax is 5.5%.
3. Write the function T(x) to represent the total cost of an item with 5.5% sales tax:
T(x) = x + (x * 0.055) = x * 1.055
4. Find the original price using the composition of functions (T of S)(x):
(T of S)(x) = T(S(x)) = T(x * 0.67) = x * 0.67 * 1.055
5. Determine the original price of the vase when the total cost is $45.95:
45.95 = x * 0.67 * 1.055
Solving this equation for x will give us the original price of the vase.