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Chapter 5: Q. 94 (page 270)

Income Taxes The function

Tg=1700+0.15g-17000

represents the 2011 federal income tax T (in dollars) due for a 鈥渕arried filing jointly鈥� filer whose modified adjusted gross income isg dollars, where17000鈮�g鈮�69000

(a) What is the domain of the function T?

(b) Given that the tax due T is an increasing linear function of modified adjusted gross income g, find the range of the function T.

(c) Find adjusted gross income g as a function of federal income tax T. What are the domain and the range of this

function?

Short Answer

Expert verified

Part (a). The domain of the function Tis17000,69000.17000,69000.

Part (b). The range of the function T is 1700,9500.

Part (c). The adjusted gross income g as a function of federal income tax T is g=Tg-17000.15+17000

The domain of and range of this function is 1700,9500and 17000,69000respectively.

Step by step solution

01

Part (a) Step 1. Find the domain

It is given that 17000鈮�g鈮�69000. So the domain of the function T is17000,69000

02

Part (b) Step 1. Find the range

Substituting g=17000in the given function, we get the lower end of the range:

Tg=1700+0.15g-17000T17000=1700+0.1517000-17000T17000=1700

Substituting g=69000in the given function, we get:

Tg=1700+0.15g-17000T69000=1700+0.1569000-17000Tg=9500

The range is given as:

1700,9500

03

Part (c) Step 1. Find the adjusted gross income g. 

The given function is Tg=1700+0.15g-17000

The adjusted gross gas the function of federal income tax T is :Tg=1700+0.15(g-17000)Tg-1700=1700-1700+0.15g-17000Subtract1700bothsidesTg-1700=0.15g-17000Tg-17000.15=0.15g-170000.15Divide0.15bothsidesTg-17000.15=g-17000g=Tg-17000.15+17000

This function is the inverse of T. So,

The domain is 1700,9500

The range is 17000,69000

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