Precalculus
Math 12
Project 1 Solutions: Some Taxing Problem Solutions

Instructions:  Write your answers to the following questions on separate paper.  Answers should be written in complete sentences.  Use a full sheet of paper (graph paper, if you like) for #1.  Label all axes and indicate coordinates of significant points.

In the US, federal taxation rates depend on “income bracket.”  The idea behind income brackets is that people who earn less money should not have to spend as high a percent of their income on taxes.  A piecewise function can be used to determine the taxes paid based on someone’s annual income. 

 

In 2003, the IRS used to use a function of the following form to determine T, the amount of taxes paid, based on I, annual taxable income.

 

1.      Sketch the graph of T = f(I) for .

2.      Use this taxation function to fill in the following table of taxes and net income vs. taxable income:

3.      Find a formula for the piecewise function f(I) which expresses federal income taxes T as a function of taxable income I.

 

 

  1. Find a formula for the piecewise function g(I) which expresses net income after taxes as a function of taxable income.  ANS:  One could simply write
    g(I) = I  f(I), or go further and simplify this as shown at right.

 

  1. Vice President Cheney earned $813,226 in 2003.  Gov. Shwarzenegger earned $26,138,000 in 2003. 
    What percent of this income would each pay in federal taxes?
    ANS:  For Cheney’s income the tax is
    90,514.50 + 0.35(813226
     311950) = $265961.10, which is 265961.1/813226 ~ 32.7%. 
               For Schwarzenegger’s income the tax is
    90,514.50 + 0.35(26138000
     311950) = $9129632, whish is 9129632/26138000 ~ 34.93%.