Problem 82 - Brief Applied Calculus

Question

The following function expresses an income tax that is \(15 \%\) for incomes below \(\$ 6000\), and otherwise is \(\$ 900\) plus \(40 \%\) of income in excess of \(\$ 6000\). \(f(x)=\left\\{\begin{array}{ll}0.15 x & \text { if } 0 \leq x<6000 \\\ 900+0.40(x-6000) & \text { if } x \geq 6000\end{array}\right.\) a. Calculate the tax on an income of \(\$ 3000\). b. Calculate the tax on an income of \(\$ 6000\). c. Calculate the tax on an income of \(\$ 10,000\). d. Graph the function.

Step-by-Step Solution

Answer

a) $450, b) $900, c) $2,500.

Step 1: Understanding the Tax Function

The function provided is a piecewise function based on income levels. For incomes below $6,000, the tax rate is 15%. For income $6,000 or more, the tax is $900 plus 40% of the income that exceeds $6,000.

Step 2: Calculate Tax for $3,000 Income

Since \(3,000 is less than \)6,000, we use the first part of the function: \[ f(x) = 0.15x \]Substitute \(3,000 into the function: \[ f(3000) = 0.15 \times 3000 = 450 \]Therefore, the tax for an income of \)3,000 is $450.

Step 3: Calculate Tax for $6,000 Income

Income is exactly \(6,000, so we use the second part of the function: \[ f(x) = 900 + 0.40(x - 6000) \]Substitute \)6,000 into the function: \[ f(6000) = 900 + 0.40(6000 - 6000) = 900 + 0 = 900 \]Thus, the tax for an income of \(6,000 is \)900.

Step 4: Calculate Tax for $10,000 Income

Since \(10,000 is greater than \)6,000, we again use the second part of the function: \[ f(x) = 900 + 0.40(x - 6000) \]Substitute \(10,000 into the function: \[ f(10000) = 900 + 0.40(10000 - 6000) = 900 + 0.40 \times 4000 = 900 + 1600 = 2500 \]Therefore, the tax for an income of \)10,000 is $2,500.

Step 5: Graph the Function

The function can be graphed as two lines. For \(0 \leq x < 6000\), the function is linear with a slope of 0.15. For \(x \geq 6000\), the function starts at \(900\) and has a steeper slope of 0.40 representing the tax increase on the excess income. The graph starts at the origin, slopes gently until \(6,000\), then sharply increases.

Key Concepts

Income Tax Calculation

Let's delve into income tax calculation using a piecewise function. The piecewise function provided helps calculate tax based on income brackets. For incomes under $6,000, the tax rate is 15%. This means the tax is computed as 15% of your total income. For instance, if your income is $3,000, your tax is found by simply multiplying 3,000 by 0.15, resulting in a tax of $450.
For incomes $6,000 or greater, the tax structure changes. You pay $900 upfront, plus 40% of the income exceeding $6,000. So, for an income of exactly $6,000, your tax is a flat $900. However, with an income of $10,000, you pay the same $900 plus 40% of the extra $4,000, resulting in a total tax of $2,500. This piecewise strategy is used by many tax systems to effectively and fairly distribute tax burdens across different income levels.
Understanding and applying these calculations helps in financial planning and ensures accurate compliance with tax requirements.

Graphing Functions

Graphing the piecewise function for tax calculation provides a visual representation of how taxes escalate with increasing income. Initially, when the income is between $0 and $6,000, the function is linear with a shallow slope of 0.15. This part of the graph represents a gradual increase in tax as income increases within this range. The graph creates a straight line originating from the origin and proceeds gently upwards.
At the $6,000 mark, there's a noticeable shift. The graph jumps to $900 due to the fixed tax amount for any income at or above this threshold. Beyond this point, the slope is steeper at 0.40, representing the larger tax rate applied to higher incomes. The intersection at $6,000 shows continuity in the piecewise function, ensuring there’s no sudden break, which aids in harmonious tax calculation increments.
Plotting this function on a graph helps easily see where the tax increases sharply, which is crucial for understanding the tax burden at different income levels.

Step-by-step Solutions

Step-by-step solutions are invaluable in understanding complex mathematical functions like the piecewise income tax function. Breaking down the problem into simple steps helps clarify the computation process.

• Start by identifying which part of the piecewise function to use, based on the income level.
• Next, substitute the income value into the appropriate equation part.
• Calculate the tax by following step-by-step arithmetic operations like multiplication and addition as illustrated.

For example, calculating the tax for a $3,000 income involves straightforward multiplication within the simpler first part of the function. In contrast, the $10,000 income requires using the second, more complex part of the function with additional steps.
This step-by-step approach ensures better understanding and enhances problem-solving skills. Each step is a building block leading to the ultimate solution, making it less daunting and more approachable for learners. It’s a methodical way to progress through the calculations while building confidence and mastery.