Chapter 1.7, Problem 29E

a.

To determine

To find: the type of function depicted by the tax rates.

Answer to Problem 29E

Piecewise function

Explanation of Solution

Given information:

The table of values given is as follows:

Income	Tax Bracket
Up to $10,000	5%
More than $10,000 but no more than $30,000	7.5%
More than $30,000	9.3%

Calculation:

Since the table shows different non-linear rates, the function of taxes due to income should be a piecewise function.

Thus, the type of function for the given situation is piecewise function.

b.

To determine

To find: the function for given condition.

Answer to Problem 29E

  $t\left(x\right)=\{\begin{array}{l}\text{5\% if \$}0<x\le \$10,000\\ \text{7}\text{.5\% if \$10,000}<x\le \$30,000\\ \text{9}\text{.3\% if }x>\$30,000\end{array}$

Explanation of Solution

Given information:

The table of values given is as follows:

Income	Tax Bracket
Up to $10,000	5%
More than $10,000 but no more than $30,000	7.5%
More than $30,000	9.3%

Let $x$ is income and $t\left(x\right)$ is tax rate.

Calculation:

Since the table shows different non-linear rates, the function of tax due to income should be a piecewise function.

So, the three parts of piecewise function will be as follows:

  $t\left(x\right)=\{\begin{array}{l}\text{5\% if \$}0<x\le \$10,000\\ \text{7}\text{.5\% if \$10,000}<x\le \$30,000\\ \text{9}\text{.3\% if }x>\$30,000\end{array}$

c.

To determine

To graph: the tax brackets for different taxable incomes

Answer to Problem 29E

  

Explanation of Solution

Given information:

The table of values given is as follows:

Income	Tax Bracket
Up to $10,000	5%
More than $10,000 but no more than $30,000	7.5%
More than $30,000	9.3%

  $t\left(x\right)=\{\begin{array}{l}\text{5\% if \$}0<x\le \$10,000\\ \text{7}\text{.5\% if \$10,000}<x\le \$30,000\\ \text{9}\text{.3\% if }x>\$30,000\end{array}$

Here, $x$ is income and $t\left(x\right)$ is tax rate.

Calculation:

The graph for the above function is as follows:

  

d.

To determine

To find: tax bracket of Ms Davis if she made $36,000 last year.

Answer to Problem 29E

Highest tax bracket

Explanation of Solution

Given information:

The table of values given is as follows:

Income	Tax Bracket
Up to $10,000	5%
More than $10,000 but no more than $30,000	7.5%
More than $30,000	9.3%

  $t\left(x\right)=\{\begin{array}{l}\text{5\% if \$}0<x\le \$10,000\\ \text{7}\text{.5\% if \$10,000}<x\le \$30,000\\ \text{9}\text{.3\% if }x>\$30,000\end{array}$

Here, $x$ is income and $t\left(x\right)$ is tax rate.

Calculation:

For $x>\$30,000$ , the tax bracket $t\left(x\right)$ is 9.3% which is the highest tax bracket. Ms Davis made $36,000 last year and hence, comes under highest tax bracket.