Q. 94 - Precalculus Enhanced with Graphing Utilities

Question

Income Taxes The function

$T\left(g\right)=1700+0.15\left(g-17000\right)$

represents the 2011 federal income tax T (in dollars) due for a “married filing jointly” filer whose modified adjusted gross income is g dollars, where $17000\le g\le 69000$

(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?

Step-by-Step Solution

Answer

Part (a). The domain of the function T  is $\left[17000,69000\right]$.$\left[17000,69000\right]$.

Part (b). The range of the function T  is $\left[1700,9500\right]$ .

Part (c).  The adjusted gross income g as a function of federal income tax T  is $g=\frac{T\left(g\right)-1700}{0.15}+17000$

The domain of and range of this function is $\left[1700,9500\right]$ and $\left[17000,69000\right]$ respectively.

Part (a) Step 1. Find the domain

It is given that $17000\le g\le 69000$. So the domain of the function T  is $\left[17000,69000\right]$

Part (b) Step 1. Find the range

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

$$\begin{gathered} T\left(g\right)=1700+0.15\left(g-17000\right) \\ T\left(17000\right)=1700+0.15\left(17000-17000\right) \\ T\left(17000\right)=1700 \end{gathered}$$

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

$$\begin{gathered} T\left(g\right)=1700+0.15\left(g-17000\right) \\ T\left(69000\right)=1700+0.15\left(69000-17000\right) \\ T\left(g\right)=9500 \end{gathered}$$

The range is given as:

$\left[1700,9500\right]$

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

The given function is $T\left(g\right)=1700+0.15\left(g-17000\right)$

The adjusted gross g as the function of federal income tax T is :$$\begin{gathered} T\left(g\right)=1700+0.15(g-17000) \\ T\left(g\right)-1700=1700-1700+0.15\left(g-17000\right) \left[Subtract 1700 both sides\right] \\ T\left(g\right)-1700=0.15\left(g-17000\right) \\ \frac{T\left(g\right)-1700}{0.15}=\frac{0.15\left(g-17000\right)}{0.15} \left[Divide 0.15 both sides\right] \\ \frac{T\left(g\right)-1700}{0.15}=g-17000 \\ g=\frac{T\left(g\right)-1700}{0.15}+17000 \end{gathered}$$

This function is the inverse of $T$. So,

The domain is $\left[1700,9500\right]$ 

The range is $\left[17000,69000\right]$