For the first 125000 of income every dollar is taxed at a ra

For the first $125,000 of income, every dollar is taxed at a rate of15% Each additional dollar over $125,000 is taxed at a rate of25%, for the next $150,000 of income. Each additional dollar over $275,000is taxed at a rate of 35%, for the next $200,000 of income Every dollar over $475,000 is taxed at a rate of 50% Find a piecewise-defined function T that specifies the yearly federal income tax for a person earning x dollars per year.

Solution

Let earning be $x

first $125,000 of income, every dollar is taxed at a rate of15%

= 0.15x ; x<= 125000

Each additional dollar over $125,000 is taxed at a rate of25%, for the next $150,000 of income

= 0.15*125000 +0.25(x- 150,000)

= 18750 +0.25x - 37500

=0.25x - 18750 ; 125000<x <= 275000

Each additional dollar over $275,000is taxed at a rate of 35%, for the next $200,000 of income:

= 0.15*125000+ 0.25(150,000) + 0.35(x-275000) ; 275000< x<= 475000

= 0.35x - 400,00 ; 275000< x<= 475000

Every dollar over $475,000 is taxed at a rate of 50%

= 0.15*125000+ 0.25(150,000) + 0.35(200000) + 0.5(x- 475000)\'

= 126250 - 237500 +0.5x

= 0.5x -111250 ; x> 475000

So, T(x) = 0.15x ; x<= 125000

= 0.25x - 18750 ; 125000<x <= 275000

=   0.35x - 400,00 ; 275000< x<= 475000

=   0.5x -111250 ; x> 475000