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 gdollars, 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?

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

Step-by-step solution

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 isrole="math" localid="1646301618846" $\left[4750,17025\right]$

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 gas the function of federal income tax T is :

$$\begin{gathered} T\left(g\right)=4750+0.25\left(g-34500\right) \\ T\left(g\right)-4750=0.25\left(g-34500\right)\left[subtracting4750bothsides\right] \\ \frac{T\left(g\right)-4750}{0.25}=g-3450\left[Divide0.25bothsides\right] \\ \frac{T\left(g\right)-4750}{0.25}+3450=g\left[Add3450bothsides\right] \\ g=\frac{T\left(g\right)-4750}{0.25}+3450 \end{gathered}$$

This function is the inverse of T. So,

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

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