Q. 41 - Precalculus Enhanced with Graphing Utilities

Question

Taxes The function $T\left(x\right) = 0.15(x - 8350) + 835$ represents the tax bill T of a single person whose adjusted gross income is x dollars for income between \(8350 and \)33,950, inclusive, in 2009. 

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 \(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 \)3707.50? 

(f) Interpret the slope. 

Step-by-Step Solution

Answer

Part (a) The domain of this linear function is $\left\{x|8350<x<33950\right\}$.

Part (b) A single filer’s tax bill is $2582.50 if adjusted gross income is $20000. 

Part (c) T is dependent variable and x is independent variable. 

Part (d) The graph is 

Part (e) A single filer’s adjusted gross income is 27500, if the tax bill is $3707.50 .

Part (f)  The slope is $m=0.15$. The tax bill of a single person increased by $0.15 for every additional dollar of taxable income.

Part (a) Step 1 Given information

The function T represents the tax bill of a single person. 

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

Where, x is adjusted gross income in dollars for income between $8350 and $33,950, inclusive, in 2009. 

Part (a) Step 2. Explanation.

In the linear function for tax bill is $T\left(x\right)=0.15(x- 8350)+835$.

x be adjusted gross income.

The income of a single person is between $8350 and $33,950.

So, the domain of this linear function is $\left\{x|8350\le x\le 33950\right\}$.

Part (b) Step 1. Explanation.

Find a single filer’s tax bill if adjusted gross income is $20000.

So we have $x=20000$.

Substitute $x=20000$ in $T\left(x\right)=0.15(x-8350)+835$$$\begin{gathered} T\left(x\right)=0.15(x-8350)+835 \\ =0.15(20000-8350)+835 \\ =2582.50 \end{gathered}$$

A single filer’s tax bill is $2582.50 if adjusted gross income is $20000. 

Part (c) Step 1. Explanation.

In linear function $T\left(x\right)=0.15(x-8350)+835$, value of T changes with x.

So, T is dependent variable and x is independent variable.

Part (d) Step 1. Explanation.

The domain of the linear functions $T\left(x\right)=0.15(x-8350)+835$ is $\left[8350, 339504\right]$.

For $x=8350$ find T.

$$\begin{gathered} T\left(x\right)=0.15(x-8350)+835 \\ =0.15(8350-8350)+835 \\ =835 \end{gathered}$$

For $x=33950$ Find T.

$$\begin{gathered} T\left(x\right)=0.15(x-8350)+835 \\ =0.15(33950-8350)+835 \\ =3840+835 \\ =4675 \end{gathered}$$

Plot the graph using points $\left(33950,4675\right)$ and $\left(8350,835\right)$.

Part (e) Step 1. Explanation.

Find a single filer’s adjusted gross income if the tax bill is $3707.50.

So we have $T\left(x\right)=3707.50$.

Find x by substituting $T\left(x\right)=3707.50$ in linear equation.

$$\begin{gathered} T\left(x\right)=0.15(x-8350)+835 \\ 3707.50=0.15(x-8350)+835 \\ 2872.5=0.15x-1252.5 \\ 4125=0.15x \\ 27500=x \end{gathered}$$ 

A single filer’s adjusted gross income is 27500, if the tax bill is $3707.50 .

Part (f) Step 1. Explanation.

Interpret the slope.

Compare the function T(x) with standard linear equation $y=mx+b$  where m is slope of the line and b is y-intercept.

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

Here $m=0.15$

Slope is average rate change. So, the tax bill of a single person increased by $0.15 for every additional dollar of taxable income.