You make a purchase at a local hardware store, but what you've bought is too big to take home in your car. for a small fee, you arrange to have the hardware store deliver your purchase for you. You pay for your purchase, plus the sales taxes, plus the fee. The taxes are 7.5% and the fee is $20.

1. Write the function t(x) for the total after taxes, on the purchase amount x. Write another function f(x) for the total, including the delivery fee on the purchase amount x.

2. Calculate and interpret f∘t and t∘f. Which results in a lower cost to you?

3. Suppose taxes by law are not to be charged on delivery fees. Which composite function must then be used? Explain you answer.

You make a purchase at a local hardware store, but what you've bought is too big to take home in your car. For a small fee, you arrange to have the hardware store deliver yourpurchase for you. You pay for your purchase, plus the sales taxes, plus the fee. The taxes are 7.5% and the fee is $20.

1. The function t(x) for the total after taxes can be written as t(x) = x + 0.075x = 1.075x. This takes the purchase amount x and adds 7.5% of that amount as taxes.

The function f(x) for the total, including the delivery fee, can be written as f(x) = t(x) + $20 = 1.075x + $20. This takes the purchase amount after taxes and adds the delivery fee of $20.

2. The composite function f∘t represents first applying the tax function (t(x)) to the purchase amount and then applying the delivery fee function (f(x)) to the result. Mathematically, it can be written as f∘t(x) = f(t(x)) = f(1.075x) = 1.075(1.075x) + $20 = 1.15625x + $20.

The composite function t∘f represents first applying the delivery fee function (f(x)) to the purchase amount and then applying the tax function (t(x)) to the result. Mathematically, it can be written as t∘f(x) = t(f(x)) = t(1.075x + $20) = 1.075(1.075x + $20) = 1.15625x + $21.5.

Comparing the two composite functions, we can see that t∘f results in a lower cost to you since it adds an additional $1.5 compared to f∘t.

3. If taxes are not charged on delivery fees by law, then the composite function t∘f must be used. This is because t∘f calculates the tax after adding the delivery fee, whereas f∘t calculates the tax before adding the delivery fee. Since taxes are not to be charged on the delivery fee, using t∘f ensures that the tax is correctly calculated based on the purchase amount after adding the delivery fee.

1. To write the function t(x) for the total after taxes on the purchase amount x, we can use the following formula:

t(x) = x + (0.075 * x)

The expression (0.075 * x) calculates the amount of taxes (7.5% of x), which is then added to the purchase amount x.

To write the function f(x) for the total including the delivery fee on the purchase amount x, we can use the following formula:

f(x) = x + (0.075 * x) + 20

Here, we add the same tax amount as in t(x) and include the delivery fee of $20.

2. To calculate f∘t, we substitute t(x) into f(x). So, f∘t can be expressed as:

f∘t = f(t(x))

Substituting t(x) into f(x):

f∘t = t(x) + (0.075 * t(x)) + 20

Simplifying the expression, we get:

f∘t = x + (0.075 * x) + 20 + (0.075 * (x + (0.075 * x)))

To calculate t∘f, we substitute f(x) into t(x). So, t∘f can be expressed as:

t∘f = t(f(x))

Substituting f(x) into t(x):

t∘f = f(x) + (0.075 * f(x))

Simplifying the expression, we get:

t∘f = (x + (0.075 * x) + 20) + (0.075 * (x + (0.075 * x) + 20))

To determine which composite function results in a lower cost to you, you need to compare the values of f∘t and t∘f for a given purchase amount x. Calculate both functions using a specific value of x, and the one with the lower cost is the preferred option.

3. If taxes are not to be charged on delivery fees, the composite function t∘f must be used. This is because t∘f calculates the total amount after applying taxes on the purchase amount and then includes the delivery fee, which aligns with the new tax exemption policy for delivery fees. On the other hand, f∘t would still include taxes on the entire purchase amount, including the delivery fee, which wouldn't comply with the new tax exemption policy.

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