Chapter 3.1, Problem 41AYU

(a)

To determine

To find: thedomain of this linear function

Answer to Problem 41AYU

The domain of given function $=\left[8350,33950\right]$

Explanation of Solution

Given:

  $T\left(x\right)=0.15\left(x-8350\right)+835$

Calculation:

Let us consider the values of x falls between $8350 and $33950 and domain of a function is the set of all values of x given.

So the domain of given function $=\left[8350,33950\right]$

Where $\left[\right]$ shows the closed interval showing that both initial and last value of interval areincluded.

Conclusion:

Therefore, the domain of given function $=\left[8350,33950\right]$

(b)

To determine

To find: the single filer’s tax bill if adjusted gross income is $20000.

Answer to Problem 41AYU

So single filer's tax bill $=\$2582.50$ .

Explanation of Solution

Calculation:

Here adjusted gross income $x=\$20,000$

So Single tax bill

  $\begin{array}{l}T\left(x\right)=0.15\left(x-8350\right)+835\hfill \\ \begin{array}{l}=0.15\left(20,000-8350\right)+835\\ =\left(0.15\times 11650\right)+835\end{array}\hfill \\ \begin{array}{l}=1747.5+835\\ =2582.50\end{array}\hfill \end{array}$

So single filer's tax bill $=\$2582.50$ .

Conclusion:

Therefore, the single filer's tax bill $=\$2582.50$ .

(c)

To determine

To describe: which variable is independent and which is dependent.

Answer to Problem 41AYU

Variable x , adjusted gross income, is independent variable andTax bill T is the dependent variable.

Explanation of Solution

Calculation:

Assume any value of variable x here from the given interval between 8350 and 33950 that is why variable x , adjusted gross income, is independent variable here for the given function.

Furthermore as the values of T would be calculated by using the given function. It is clear that the values of T is totally depends on the values of x , to be put in the function.

So here Tax bill T is the dependent variable.

Conclusion:

Therefore, variable x , adjusted gross income, is independent variable andTax bill T is the dependent variable.

(d)

To determine

To graph: the linear function over the domain specified in part (a).

Answer to Problem 41AYU

The graph of given linear function are $\left(8350,835\right)$ and $\left(8350,835\right)$ .

Explanation of Solution

Calculation:

Now when $x=8350$

  $\begin{array}{l}T\left(x\right)=0.15\left(x-8350\right)+835\hfill \\ \begin{array}{l}=0.15\left(8350-8350\right)+835\\ =0.15\times 0+835\\ =835\end{array}\hfill \end{array}$

And when $x=\$33,950$

  $\begin{array}{l}T\left(x\right)=0.15\left(x-8350\right)+835\hfill \\ \begin{array}{l}=0.15\left(33,950-8350\right)+835\\ =\left(0.15\times 25600\right)+835\\ =3840+835\end{array}\hfill \\ =4675\hfill \end{array}$

So two corresponding coordinates of form $\left(x,T\left(x\right)\right)$ on the graph of given linear function are $\left(8350,835\right)$ and $\left(8350,835\right)$ .

  

Conclusion:

Therefore, the graph of given linear function are $\left(8350,835\right)$ and $\left(8350,835\right)$ .

(e)

To determine

To find: the single filer’s adjusted gross income if the tax bill is $3707.50.

Answer to Problem 41AYU

The required single filer's adjusted gross income =$27,500.

Explanation of Solution

Calculation:

Now tax bill $T=\$3707.50$

So on plug in this value of T in given function,

  $\begin{array}{l}T\left(x\right)=0.15\left(x-8350\right)+835\hfill \\ \begin{array}{l}3707.50=0.15\left(x-8350\right)+835\\ 3707.50-835=0.15\left(x-8350\right)+835-835\end{array}\hfill \\ 2872.5=0.15\left(x-8350\right)\hfill \end{array}$

Now on dividing both sides by 0.15,

  $\begin{array}{l}\begin{array}{l}\frac{2872.5}{0.15}=\frac{0.15\left(x-8350\right)}{0.15}\\ 19150=x-8350\end{array}\hfill \\ 19150+8350=x-8350+8350\hfill \\ x=27,500\hfill \end{array}$

So the required single filer's adjusted gross income =$27,500

Conclusion:

Therefore, the required single filer's adjusted gross income =$27,500.