Chapter 3.1, Problem 41AYU

Taxes The function $T\left(x\right)=0.15(x-9225)+922.50$ represents the tax bill $T$ of a single person whose adjusted gross income is $x$ dollars for income between $\$9225$ and $\$\mathrm{37,450}$, inclusive, in 2015.

Source: Internal Revenue Service

a. What is the domain of this linear function?

b. What is a single filer's tax bill if adjusted gross income is $\$\mathrm{20,000}$ ?

c. Which variable is independent and which is dependent?

d. Graph the linear function over the domain specified in part (a).

e. What is a single filer’s adjusted gross income if the tax bill is $\$3663.75$?

f. Interpret the slope.

To determine

To calculate:

a. What is the domain of this linear function?

b. What is a single filer’s tax bill if adjusted gross income is $\$20,000$?

c. Which variable is independent and which is dependent?

d. Graph the linear function over the domain specified in part (a..

e. What is a single filer’s adjusted gross income if the tax bill is $\$3663.75$?

f. Interpret the slope.

Answer to Problem 41AYU

Solution:

a. The domain of the given function is $(0,\infty )$.

b. $\$2538.75$ is a single filer’s tax bill if adjusted gross income is $\$20,000$.

c. The graph is shown below:

d. $T\left(x\right)$ is dependent on the adjusted gross income $x$ which is independent of any other variable and there are no dependent variables in this function.

e. When the tax bill $\$3663.75$, the adjusted gross income is $\$27500$.

f. When the adjusted gross income increases then the tax bill also increases.

Explanation of Solution

Given:

The given tax bill function is $T(x)=0.15(x-9225)+922.50$.

The relations of a tax bill function $T$ of a single person whose adjusted gross income is $x$ dollars for income between $\$9225$ and $\$37450$, inclusive, in 2015.

Calculation:

a. We can see that the value of $x$ in the given function can be any number between 0 and $\infty$. Therefore the domain of the given function is $(0,\infty )$.

b. If $x=20000$, we get

$T(x)=0.15(20000-9225)+922.50$

$\Rightarrow T(x)=2538.75$

c. Here, we can see that the tax bill $T\left(x\right)$ is dependent on the adjusted gross income $x$ which is independent of any other variable and there are no dependent variables in this function.

d. The graph of the given function in the domain $(0,\infty )$ is

e. If $T(x)=3663.75$, we get

$3663.75=0.15(x-9225)+922.50$

$\Rightarrow x=\frac{3663.75-922.5}{0.15}+9225$

$\Rightarrow x=27500$

Therefore, when the tax bill is $\$3663.75$, the adjusted gross income is $\$27500$.

f. Here, we can see that the slope is positive; therefore the function is an increasing function, i.e. when the adjusted gross income increases then the tax bill also increases.