Q. 93 - Precalculus Enhanced with Graphing Utilities

Question

Income Taxes. The function

$T\left(g\right)=4750+0.25\left(g-34500\right)$

represents the 2011 federal income tax T (in dollars) due for a “single” filer whose modified adjusted gross income is g dollars, where $34500\le g\le 83600$.

(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 $\left[34500,83600\right]$.

Part (b). The range of the function T  is $\left[4750,17025\right]$.

Part (c).  The adjusted gross income g as a function of federal income tax T  is $g=\frac{T\left(g\right)-4750}{0.25}+3450$ 

The domain of and range of this function is $\left[4750,17025\right]$ and $\left[34500,83600\right]$.

Part (a) Step 1. Find the domain

 is given that $34500\le x\le 83600$.So. the domain of the function T  is $\left[34500,83600\right]$

Part (b) Step 1. Find the range

Substituting $g=34500$ in the given function, we get the lower end of the range:  

$$\begin{gathered} T\left(g\right)=4750+0.25\left(g-34500\right) \\ T\left(g\right)=4750+0.25\left(34500-34500\right) \\ T\left(g\right)=4750 \end{gathered}$$

Substituting $g=83600$ in the given function, we get:

$$\begin{gathered} T\left(g\right)=4750+0.25\left(g-34500\right) \\ T\left(g\right)=4750+0.25\left(83600-34500\right) \\ T\left(g\right)=17025 \end{gathered}$$

Thus, the range is  $\left[4750,17025\right]$

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

The given function is $T\left(g\right)=4750+0.25\left(g-34500\right)$

The adjusted gross g as the function of federal income tax T is : 

$$\begin{gathered} T\left(g\right)=4750+0.25\left(g-34500\right) \\ T\left(g\right)-4750=0.25\left(g-34500\right) \left[subtracting 4750 both sides\right] \\ \frac{T\left(g\right)-4750}{0.25}=g-3450 \left[Divide 0.25 both sides\right] \\ \frac{T\left(g\right)-4750}{0.25}+3450=g \left[Add 3450 both sides\right] \\ g=\frac{T\left(g\right)-4750}{0.25}+3450 \end{gathered}$$

This function is the inverse of T. So, 

The domain of the function is $\left[4750,17025\right]$

The range of the function is $\left[34500,83600\right]$