Computing Income Tax The tax that you pay to the federal...

Computing Income Tax The tax that you pay to the federal government is a percentage of your taxable income, which is what remains of your gross income after you subtract your allowed deductions. In a recent year, there were five rates or brackets for a single taxpayer, as shown in Table $1 .$ So, if you are single and your taxable income was less than $\$ 27,050$, your tax is your taxable income times $15 \%$ (.15). The maximum amount of tax that you will pay on your income in this first bracket is $15 \%$ of $\$ 27,050$, or $(.15) \times 27,050=4057.50$ dollars. If your taxable income is more than $\$ 27,050$ but less than $\$ 65,550$, your tax is $\$ 4057.50$ plus $27.5 \%$ of the amount in excess of $\$ 27,050 .$ So, for example, if your taxable income is $\$ 50,000$, your $\operatorname{tax}$ is $\$ 4057.5+.275(50,000-27,050)=4057.5+.275 \times$ $22,950=\$ 10,368.75 .$ Let $x$ denote your taxable income and $T(x)$ your tax. (a) Find a formula for $T(x)$ if $x$ is not over $\$ 136,750$. (b) Plot the graph of $T(x)$ for $0 \leq x \leq 136,750$. (c) Find the maximum amount of tax that you will pay on the portion of your income in the second tax bracket. Express this amount as a difference between two values of $T(x)$.

Computing Income Tax The tax that you pay to the federal government is a percentage of your taxable income, which is what remains of your gross income after you subtract your allowed deductions. In a recent year, there were five rates or brackets for a single taxpayer, as shown in Table $1 .$

So, if you are single and your taxable income was less than $\$ 27,050$, your tax is your taxable income times $15 \%$ (.15). The maximum amount of tax that you will pay on your income in this first bracket is $15 \%$ of $\$ 27,050$, or $(.15) \times 27,050=4057.50$ dollars. If your taxable income is more than $\$ 27,050$ but less than $\$ 65,550$, your tax is $\$ 4057.50$ plus $27.5 \%$ of the amount in excess of $\$ 27,050 .$ So, for example, if your taxable income is $\$ 50,000$, your $\operatorname{tax}$ is $\$ 4057.5+.275(50,000-27,050)=4057.5+.275 \times$
$22,950=\$ 10,368.75 .$ Let $x$ denote your taxable income and $T(x)$ your tax.
(a) Find a formula for $T(x)$ if $x$ is not over $\$ 136,750$.
(b) Plot the graph of $T(x)$ for $0 \leq x \leq 136,750$.
(c) Find the maximum amount of tax that you will pay on the portion of your income in the second tax bracket. Express this amount as a difference between two values of $T(x)$.

Key Concepts

Maximum Tax within a Bracket

The maximum tax within a tax bracket refers to the highest amount of tax incurred from the income within that specific bracket. It is often calculated as the difference between the tax calculated at the upper limit and the lower limit of the bracket, helping to delineate the cost of income within that range.

Graphical Representation of Piecewise Functions

Graphing piecewise functions involves plotting each segment defined by the tax brackets over its respective interval. This visual representation helps in understanding how the tax liability changes as taxable income increases and transitions from one bracket to another.

Marginal Tax Rates

Marginal tax rates apply to income within specific brackets; they indicate the percentage of tax applied to each additional dollar earned within that bracket. This system ensures that higher portions of income are taxed at higher rates, while only the income above certain thresholds is taxed at the higher rate.

Piecewise Functions

Piecewise functions are mathematical functions defined by different expressions over various intervals of the independent variable. In the context of income tax, they are used to model how tax liability is calculated differently depending on which tax bracket the income falls into.

Allowed Deductions

Allowed deductions are specific expenses or amounts that can be subtracted from gross income to reduce the overall taxable income. This process helps in lowering the tax liability by ensuring that tax is only paid on income that exceeds these deductions.

Taxable Income

Taxable income represents the portion of an individual's income that is subject to taxation after subtracting all allowed deductions from the gross income. It is the base amount on which the tax is calculated and is crucial for determining how much tax an individual owes.

Tax Brackets

Tax brackets divide income ranges into segments where each segment is taxed at a specific rate. As income increases and falls into higher brackets, the amount of tax owed increases at varying marginal rates, reflecting a progressive tax system.

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