Problem 90 - College Algebra

Question

During a particular year, the taxes owed, \(T(x),\) in dollars, filing separately with an adjusted gross income of \(x\) dollars is given by the piecewise function $$ T(x)=\left\\{\begin{array}{ll} 0.15 x & \text { if } 0 \leq x<17,900 \\ 0.28(x-17,900)+2685 & \text { if } 17,900 \leq x<43,250 \\ 0.31(x-43,250)+9783 & \text { if } x \geq 43,250 \end{array}\right. $$ In Exercises \(89-90,\) use this function to find and interpret each of the following. 90\. \(T(70,000)\)

Step-by-Step Solution

Answer

Given this piecewise function, \(T(70,000)\) equals \$16,591. This means that a person with an income of \$70,000 would owe \$16,591 in taxes.

Step 1: Identify the Relevant Piecewise Function

First, determine which equation within the piecewise function applies. The income $70,000 falls within the range of $43,250 and infinity. Thus, the equation to use is the third one: \( T(x) = 0.31(x - 43,250) + 9783 \)

Step 2: Substitute for x

Substitute \$70,000 for x in the equation you've identified: \( T(70,000) = 0.31(70,000 - 43,250) + 9783 \)

Step 3: Simplify

Now, simplify the equation. Start by calculating the term within the parenthesis: \( T(70,000) = 0.31(26,750) + 9783 \). Finally solving this you have to calculate 0.31 times 26,750 and add it to 9783.

Key Concepts

Piecewise Function in Algebra

A piecewise function in algebra is essentially a mathematical function defined by multiple sub-functions, each of which applies to a certain interval within the whole function's domain.

Imagine you're reading a cookbook, and for the recipe, different ingredients and quantities are needed depending on the number of guests. A piecewise function works similarly; it has different 'recipes' for inputs that fall into different ranges.

For example, in the given tax function, there are three 'recipes' or segments: one for incomes below \(17,900, another for incomes between \)17,900 and \(43,250, and a third one for incomes above \)43,250. The function 'switches' between these intervals based on the value of the input, your income, in this case.

Tax Function Calculation

When it comes to tax functions, calculating the amount of taxes owed often involves a piecewise function. These functions reflect the real-world situation where different income ranges are taxed at different rates.

For the exercise at hand, the tax function is a clear example of a piecewise function where each segment corresponds to a tax bracket. To calculate the tax owed on an income of $70,000, we locate which 'piece' or 'segment' of the function applies to this income bracket and use it for our calculation.

This mathematical approach allows a realistic tax system to be modeled, where the effective tax rate increments with higher income—often referred to as a progressive tax system.

Interpreting Mathematical Functions

Functions are like special rules that tell you what output you'll get for any given input. In our tax calculation, the function tells us what amount is owed in taxes for any level of income.

Interpreting such mathematical functions means not just calculating the figures correctly, but understanding what they represent in the real world. It's like when you read a weather chart; the information is more than numbers—it tells a story about what to expect when you step outside.

Solving Algebraic Equations

When solving algebraic equations, especially in complex scenarios like a piecewise function, it's important to follow systematic steps. Let's take a closer look at the process in our tax function example:

• Step 1: Identify which part of the piecewise function to use based on the given input.
• Step 2: Substitute the given input into the equation you've identified.
• Step 3: Execute the calculation as you would with a regular algebraic equation, by simplifying step by step and performing the arithmetic operations.

Following these systematic steps helps to avoid mistakes and ensures the algebraic problems are solved accurately, step by step.