I am having trouble with piecewise word problems.. I done the first two questions but don't understand the third one which is..

Question

The table below shows the 2002 federal income tax rates for a single person

taxable income tax

not over $6000 10% of income

not over $6000, $600+15% of amount
but not over $27, 950 over $6000

over $27, 950 but not $3892.50+ 27% of
over $67, 700 amount over $27,950

over $67, 700 $14,625+30% of
but not over $141, 250 amount over $67,700

over $141, 250 but not $36, 690+ 35% of
over $307, 050 amount over $141,250

over $307, 050 $94, 720+ 38.6% of
amount over $307, 050

Write a precise-defined function T such that T(x) is the tax due on a taxable income of x dollars. What is the domain of the function?

Find T(24,000), T(35,000), T(100,000)

Find the amount of taxable income if the following was due

$4, 446
$16, 815

Answer 1

note that at each break point, the base amount is the maximum tax of the previous bracket.

That means that for income
<= 6000 max tax is 600
<= 27950 max tax is 3892
<= 67700 max tax is 3892.50

So, tax of 4446 means that income was between 27950 and 67700

3892 + .27(x-27950) = 4446
x = 30001.90

move on to the next brackets as needed for the higher tax amount.