# Tax brackets

Suppose tax brackets are as follows: for taxable income between $0 to$32000, the tax rate is 15%; for taxable income between $32000 to$64000, the tax rate is 22%; for taxable income between $64000 and$100000, the tax rate is 25%, for income above $100000, the tax rate is 29%. a) If x is the taxable income, and T(x) is the tax, find a formula for tax up to$100000.
b) Plot the graph of taxation for taxable income up to $100000. C) Write, in terms of T(x), the max taxation on the income between$64000 and $100000. Answer a) Because the taxation is separated into brackets, we have to separate the answer into "brackets" as well. First of all, for income below$32000, we pay 15% tax. So if income is x < $32000, then tax T(x) = 0.15x. At x =$32000, tax is ${\displaystyle 32000\times 0.15=\4800}$. Now, if taxable income is between $32000 and$64000, only the portion of income above $32000 has to pay 22% of tax, (and the portion below$32000 would pay 15%, which amounts to $4800), then the tax ${\displaystyle T(x)=4800+(x-32000)\times 0.22}$. At x = 64000, tax = 4800 + (tax on portion above$32000). The tax on portion over $32000 is ${\displaystyle (\64000-\32000)\times 0.22=\7040}$. So, at income x = 64000, tax = 4800 + 7040 = 11840. Now, turning to income > 64000, the tax ${\displaystyle T(x)=11840+(x-64000)\times 0.25}$. b) If T(x) is on the vertical axis and x is on the horizontal axis, then we plot the following points (0, 0), (32000, 4800), (64000, 11840) and (100000, t), where t = tax you pay at$100000. Plug $100000 into formula ${\displaystyle T(x)=11840+(x-64000)\times 0.25=20840}$. That is t = 20840. c) The max tax between$64000 and \$100000 is just T(100000) - T(64000) = 20840 - 11840 = 9000.