3. Joyce 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.

An auto mechanic charges $35 per hour for labor plus $45 for parts for a tune up on six-cylinder car. The cost of a tune up function determined by the formula C(t)=35t+45. The amount of time for the tune up is determined by a formula given in the manual, A(x)=0.5x, where x is a numerical code for the particle make of car.Write a formula for the cost of tune up in terms of the numerical code. If the numerical code for certain car is 3, determine the final cost of the tune up.

Thank you for all the help

To solve these questions, we will use composite functions and some basic algebra. Let's go step by step for each question:

1. Finding the composite function (T of S)(x) and the final cost of the jeans:
To find the total cost of an item with 7% sales tax, we need to add the cost of the item and the sales tax. The formula for the total cost with sales tax is: T(x) = x + 7% of x = x + 0.07x.
Now, to find the composite function (T of S)(x), we need to apply the discount before applying the sales tax. The formula becomes: (T of S)(x) = T(S(x)).
Since the jeans are on sale at a 20% discount, the discounted price is 80% of the original price: S(x) = 80% of x = 0.8x.
Substituting this into the formula for T(x), we get: (T of S)(x) = T(0.8x) = 0.8x + 0.07*(0.8x).
Simplifying this expression, we get: (T of S)(x) = 0.8x + 0.056x = 0.856x.
Therefore, the composite function (T of S)(x) is 0.856x.
To find the final cost of the jeans, substitute the original price of the jeans into the composite function: (T of S)(39.99) = 0.856 * 39.99. Calculate this to find the final cost of the jeans.

2. Determining the original price of the vase using the composition of functions:
The original price of the vase can be determined by reversing the steps we used to find the final cost. Let's call the original price of the vase 'P'.
The sales price of the vase after a 33% discount is 67% of the original price: S(P) = 67% of P = 0.67P.
Adding the 5.5% sales tax, the final cost of the vase is: T(S(P)) = 0.67P + 5.5% of (0.67P) = P.
Set that equation equal to the given final cost of the vase ($45.95), and solve for P to find the original price of the vase.

3. Writing a formula for the cost of a tune-up in terms of the numerical code, and finding the final cost:
The formula for the cost of a tune-up is given as: C(t) = 35t + 45, where t represents the number of labor hours.
However, we need to express the cost in terms of the numerical code x for the vehicle make.
Given that the time for the tune-up is determined by the formula A(x) = 0.5x, we can substitute this into the formula for the cost: C(A(x)) = 35 * A(x) + 45.
Substitute A(x) = 0.5x into the formula to get the cost in terms of the numerical code x.
Finally, substitute the given numerical code (3) into the formula to find the final cost of the tune-up.

I hope this explanation helps you understand how to solve these problems. Let me know if you have any further questions!