Arizona taxes problem.

Question

The following is from the actual Arizona state tax table for 2020 taxes.

A single person whose taxable income is between $0-$27,272 is charged 2.59% in tax.

A single person whose taxable income is between $27,273-$54,544 is charged 2.59% on the first $27,272, and then they are taxed 3.34% on their amount over $27,272.

For example, an instructor with a taxable income of $40,000 would get taxed $1131.46.

A) Write a piecewise-linear function to represent the total amount of AZ taxes owed for a single taxpayer as a function of taxable income, for taxable incomes up to $54,544.

B) Using your function, what is the amount of state taxes owed for a single person whose taxable income is $20,000? $50,000?

Answer 1

To representing the total taxes owed using a piecewise function, we will need to create two different equations: one calculating taxes if their income is under $27,273, and one if their income is between $27,273 and $54,544. The second equation will be more complicated because we will need to break the income into two parts. However, to make the function work properly and calculate totals, we cannot simply make it into a separate equation altogether. A good piecewise function for this would look like this:

____

| (0.0259)x if 0 ≤ x ≤ 27,272

ƒ(x) = { (0.0334)(x - 27,272) + 0.0259(27,272) if 27,273 ≤ x ≤ 54,544

|____

The second piece of the second equation is just a number: 0.0259(27,272) = 706.3448. This is because if the person has income above 27,272, we already know they will have the first 27,272 taxed at 2.59%. The rest of what they are owed is calculated in the first piece of the second equation. We subtract 27,272 from their income because that will give us the amount of money they make ABOVE that amount. We then multiple that by 3.34%. We add them together.

So, if their taxable income is 20,000: we look at our equation, see that x is between 0 and 27,272, so we only look at the first equation. They owe $518.

If their taxable income is 50,000: we see that the second equation is appropriate now. We get (0.0334)(50,000 - 27,272) + 0.0259(27,272) = $1465.46