Problem 108 - Precalculus Essentials

Question

Here is the Federal Tax Rate Schedule \(X\) that specifies the tax owed by a single taxpayer for a recent year. (TABLE CANNOT COPY) The preceding tax table can be modeled by a piecewise function, where \(x\) represents the taxable income of a single taxpayer and \(T(x)\) is the tax owed: $$T(x)=\left\\{\begin{array}{ccc} 0.10 x & \text { if } & 0379,150 \end{array}\right.$$ Use this information to solve. Find and interpret \(T(50,000)\)

Step-by-Step Solution

Answer

The tax owed by a single taxpayer with a taxable income of \$50,000 for the given year is \$8625.00.

Step 1: Identify the correct function

Check each interval of the piecewise function to identify which part of the function applies to \(x = 50,000\). According to the given conditions, \(50,000\) falls in the range \(\$34,500 < x \leq \$83,600\). Therefore, we will use the function \(T(x) = 4750.00 + 0.25(x-34500)\) for the given calculation.

Step 2: Substitute x-values into the selected function

For \(x = 50,000\), substitute this value of x into the selected function. It will be \(T(50,000) = 4750.00 + 0.25(50,000 - 34,500)\).

Step 3: Compute the result

Calculate the solution. This calculation will give \(T(50,000) = 4750.00 + 0.25(15,500)\), which equals \$8625.00.

Step 4: Interpret the result

The value obtained i.e. \$8625.00 is the tax owed by a single taxpayer with an income of \$50,000. Thus, if a single taxpayer has a taxable income of \$50,000, they will owe \$8625.00 in taxes for the given year as per the provided tax model.

Key Concepts

Federal Tax Rate

Understanding the federal tax rate system is crucial for taxpayers. Federal tax rates are designed to ensure that taxes are levied fairly based on the income of individuals. The tax rate schedule consists of brackets, each with a specified rate that applies to income within that range. This means that as one's income increases, they may move into higher tax brackets.
This system is progressive, meaning that it aims to have taxpayers with higher incomes pay a larger percentage of their income in taxes, while those with lower incomes pay a smaller percentage.
In practical terms, each tax bracket is associated with a specific rate applied to income that falls within its limits. This results in a piecewise function, where different rules apply depending on which income bracket a person’s taxable income falls into.

• The lowest income bracket is taxed at the lowest rate.
• As income increases, it may be subject to several different rates, one for each bracket it falls under.

Taxable Income

Taxable income is the portion of an individual's or a company's income that is subject to taxation. It includes wages, salaries, bonuses, and other forms of earnings minus deductions and exemptions provided by tax law.
Calculating taxable income correctly is important because it determines how much an individual owes in taxes. Tax brackets apply to taxable income, not total income, which makes it important to understand what constitutes taxable income.

• Taxable income reflects income after all applicable deductions.
• This income is used within the piecewise function to determine the tax owed.
• Understanding taxable income helps taxpayers estimate their tax obligations and manage their finances effectively.

Function Evaluation

Function evaluation refers to the process of substituting a specific input value into a function to determine the corresponding output. In the case of calculating federal taxes using a piecewise function, function evaluation helps determine the exact amount of tax owed based on taxable income.
To evaluate a function properly, particularly in a piecewise format:

• Identify which segment or piece of the function applies to the input value (i.e., taxable income).
• Substitute the input value into the correct equation.
• Perform the necessary mathematical calculations to obtain the result.

For instance, with a taxable income of $50,000, you identify that this amount falls within the appropriate bracket (e.g., $34,500 to $83,600). By substituting $50,000 into the corresponding piece of the function, you compute the tax owed at this income level.