Chapter 5.2, Problem 94AYU

(a)

To determine

The domain of a function $T$, where $T(g)=1905+0.12\left(g-19,050\right)$ represents the $2018$ federal income tax $T$ (in dollars) due for a “married filing jointly” filer, whose modified adjusted gross income is $g$ dollars, where $19,050\le g\le 77,400$ .

Answer to Problem 94AYU

Solution:

The domain of function $T$ is $\left[19,050,77,400\right]$ .

Explanation of Solution

Given information:

  $T(g)=1905+0.12\left(g-19,050\right)$ represents the $2018$ federal income tax $T$ (in dollars) due for a “married filing jointly” filer, whose modified adjusted gross income is $g$ dollars, where $19,050\le g\le 77,400$

The values of $g$ works as an input to the function $T$

Therefore, the domain of $T$ is $\left[19,050,77,400\right]$

(b)

To determine

The range of the function $T$ , where $T$ is the tax due is an increasing linear function of modified adjusted gross income $g$ .

Answer to Problem 94AYU

Solution:

The range of the function $T$ is $\left[1905,8907\right]$ .

Explanation of Solution

Given information:

  $T(g)=1905+0.12\left(g-19,050\right)$ represents the $2018$ federal income tax $T$ (in dollars) due for a “married filing jointly” filer, whose modified adjusted gross income is $g$ dollars, where $19,050\le g\le 77,400$ .

The tax due $T$ is an increasing linear function of modified adjusted gross income $g$ .

Therefore, the range of $T(g)$ will have the lowest value at $g=19,050$ .

Thus, substitute $g=19,050$ in equation of $T(g)$

  $\Rightarrow T(19,050)=1905+0.12\left(19,050-19,050\right)$

  $\Rightarrow T(19,050)=1905+0.12\left(0\right)$

  $\Rightarrow T(19,050)=1905+0$

  $\Rightarrow T(19,050)=1905$

The maximum value of $T(g)$ occurs at $g=77,400$

Thus, substitute $g=77,400$ in the equation of $T(g)$

  $\Rightarrow T(77,400)=1905+0.12\left(77,400-19,050\right)$

  $\Rightarrow T(77,400)=1905+0.12\left(58350\right)$

  $\Rightarrow T(77,400)=1905+7002$

  $\Rightarrow T(77,400)=8907$

Thus, the range of $T(g)$ is $\left[1905,8907\right]$

(c)

To determine

The adjusted gross income $g$ as a function of federal income tax $T$ . Determine the domain and the range of this function.

Answer to Problem 94AYU

Solution:

1) The adjusted gross income function is $g\left(T\right)=\frac{T+381}{0.12}$ .

2) The domain of $T(g)$

  $\{T(g)|1905\le T(g)\le 8907\}$ and range is $\left\{g|19,050\le g\le 77,400\right\}$ .

Explanation of Solution

Given information:

  $T(g)=1905+0.12\left(g-19,050\right)$ represents the $2018$ federal income tax $T$ (in dollars) due for a “married filing jointly” filer, whose modified adjusted gross income is $g$ dollars, where $19,050\le g\le 77,400$ .

Let $T(g)=1905+0.12\left(g-19,050\right)$

To find gross income function $g$ , solve the equation $T(g)=1905+0.12\left(g-19,050\right)$ for variable $g$ .

  $\Rightarrow T=1905+0.12g-0.12\cdot \left(19,050\right)$

  $\Rightarrow T=1905+0.12g-2286$

  $\Rightarrow T=0.12g-381$

  $\Rightarrow 0.12g=T+381$

  $\Rightarrow g=\frac{T+381}{0.12}$

Functions $T\left(g\right)$ and $g\left(T\right)$ are inverses of each other.

From part (a) and (b), the domain of $T\left(g\right)$ is $\left[19,050,77,400\right]$ and the range of $T(g)$ is $\left[1905,8907\right]$ .

Thus, domain of $g\left(T\right)$ = range of $T\left(g\right)$= $\left[1905,8907\right]$ and range of $g\left(T\right)$ = domain of $T(g)$ = $\left[19,050,77,400\right]$ .