(Federal Income Taxes and Piecewise Functions MC)

Use the marginal tax rate chart to answer the question.

Tax Bracket Marginal Tax Rate
$0–$10,275 10%
$10,276–$41,175 12%
$41,176–$89,075 22%
$89,076–$170,050 24%
$170,051–$215,950 32%
$215,951–$539,900 35%
> $539,901 37%

Part A: Demonstrate how to calculate the amount of taxes owed on a taxable income of $39,000. (4 points)

Part B: The piecewise function represents the amount of taxes owed, f(x), as a function of the taxable income, x. Using the same taxable income from Part A, demonstrate how to calculate the amount of taxes owed using the piecewise function. (4 points)

Part C: Compare the piecewise function to the marginal tax rate chart. How is the marginal tax rate chart represented in the piecewise function? (2 points)

You have a marginal rate tax chart and a piecewise function for computing the tax amount. You want to use these to compute taxes on $39,000, and compare the methods.

Part A:

The tax on $39,000 is computed in two parts: (a) the tax on the first $10,275, and (b) the tax on the amount over $10,275. Different rates apply to these amounts.

tax = 0.10×10275 +0.12×(39000 -10275) = 1027.50 +3447.00 = 4475.50

Part B:

The piecewise function computes the tax directly, based on the income value. For $39,000, the tax is ...

f(39000) = 0.12(39000) -205.50 = 4680 -205.50 = 4475.50

Part C:

For the income value used in this problem, and for income values up to 89075, the tax rate chart and the piecewise function will give the same result.

If we assume the piecewise function is supposed to match the tax rate chart, then the function shown in the problem statement has several errors. The correct function would be ...

[tex]f(x)=\begin{cases}0.10x&x\le10275\\0.12x-205.50&10275 < x\le41175\\0.22x-4323.00&41175 < x\le89075\\0.24x-6104.50&89075 < x\le170050\\0.32x-19708.50&170050 < x\le215950\\0.35x-26187.00&215950 < x\le539900\\0.37x-36985.00&539900 < x\end{cases}[/tex]

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Additional comment

You can see what the subtracted values are supposed to be by computing the tax using the chart. For example, for an income (x) between 89075 and 170050, the marginal rate chart tells you ...

tax = 0.10(10275) +0.12(41175 -10275) +0.22(89075 -41175) +0.24(x -89075)

Rearranging, this is equivalent to ...

tax = 10275(0.10 -0.12) +41175(0.12 -0.22) +89075(0.22-0.24) +0.24x

= -205.50 -4117.50 -1781.50 +0.24x

= 0.24x -6104.50 . . . . . . . compare to the piecewise function

Tax marginal rate charts like the one here consistently leave gaps between the domains. For example, the tax is undefined for incomes between $10275 and $10276. In fact, the tax on 10276 is $1027.50 plus 0.12×$1.00 = $1027.62. That is, the 12% rate applies to every dollar over 10275.

Perhaps this is done because both income and tax amounts are rounded to the nearest dollar in practice. However, it leaves one wondering exactly how the tax is supposed to be computed for amounts beyond the first bracket.