Problem 81 - Brief Applied Calculus

Question

The following function expresses an income tax that is \(10 \%\) for incomes below \(\$ 5000\), and otherwise is \(\$ 500\) plus \(30 \%\) of income in excess of \(\$ 5000\). \(f(x)=\left\\{\begin{array}{ll}0.10 x & \text { if } 0 \leq x<5000 \\\ 500+0.30(x-5000) & \text { if } x \geq 5000\end{array}\right.\) a. Calculate the tax on an income of \(\$ 3000\). b. Calculate the tax on an income of \(\$ 5000\). c. Calculate the tax on an income of \(\$ 10,000\). d. Graph the function.

Step-by-Step Solution

Answer

Tax on $3000 is $300, on $5000 is $500, and on $10000 is $2000.

Step 1: Understanding the piecewise function

The income tax function is defined in two parts: for incomes below $5000, the tax is $0.10 of the income. For incomes $5000 and above, the tax is a flat $500 plus 30% of the income exceeding $5000.

Step 2: Calculate tax on $3000

For an income of \(3000, use the first part of the function since \)3000 < \(5000: \[ f(x) = 0.10 \times 3000 \] Compute it: \[ f(x) = 300 \] So, the tax on an income of \)3000 is $300.

Step 3: Calculate tax on $5000

For an income of \(5000, use the second part of the function: \[ f(x) = 500 + 0.30 \times (5000 - 5000) \] This simplifies to: \[ f(x) = 500 + 0 = 500 \] Hence, the tax on an income of \)5000 is $500.

Step 4: Calculate tax on $10000

For an income of \(10000, use the second part of the function as \)10000 \geq 5000: \[ f(x) = 500 + 0.30 \times (10000 - 5000) \] Calculate the excess amount: \[ 0.30 \times 5000 = 1500 \] Thus, \[ f(x) = 500 + 1500 = 2000 \] Therefore, the tax on an income of \(10000 is \)2000.

Step 5: Graph the piecewise function

To graph the function, note that there are two segments: - For \(0 \leq x < 5000, the line is \)f(x) = 0.10x\(; this is a straight line passing through the origin with a slope of 0.10. - For \)x \geq 5000, the line is \(f(x) = 500 + 0.30(x - 5000)\); this is a line that starts at point (5000, 500) and has a slope of 0.30. Plot both segments on the same graph with corresponding starting points and slopes.

Key Concepts

Income Tax Calculation

The concept of **income tax calculation** can be understood well through piecewise functions. In this case, our function defines the tax a person needs to pay based on their income bracket. For incomes below \(5000, the tax is 10% of the income, which can be expressed mathematically as \(0.10x\) where \(x\) is the income. This is a consistent percentage calculation, making it straightforward to compute.

For incomes that are \)5000 or more, the tax calculation becomes a bit more complex. The tax here is a base \(500, plus an additional 30% on the amount that exceeds \)5000. This part of the function is represented as \(500 + 0.30(x - 5000)\). Understanding how each component of the function works allows us to find the amount of tax owed simply by substituting the income into the appropriate part of the function based on income amount.

Graphing Functions

When **graphing piecewise functions**, it's essential to look at each segment separately. The given income tax function consists of two segments, each with its own rule.

• For the first segment, which is applicable for incomes below \(5000, you graph the function \(f(x) = 0.10x\). This segment is a straight line that passes through the origin with a slope of 0.10, representing the 10% tax on income.
• For the second segment, applicable for incomes of \)5000 and above, you graph \(f(x) = 500 + 0.30(x - 5000)\). This line starts at the point (5000, 500) and has a steeper slope of 0.30, showing the increased tax rate.

Breaking each part down visually on a graph can aid in understanding how the tax changes as a function of income, capturing the transition points and changes in slope-contribution across different income ranges.

Slope of a Line

The **slope of a line** is a critical concept when understanding how variables relate to each other in linear equations. In the context of the income tax function, the 'slope' represents how quickly the tax amount increases as the income increases.

For the first part of our piecewise function, the slope is 0.10. This indicates that for every additional dollar of income below $5000, the tax increases by 10 cents. This represents a steady, linear increase at a constant rate.

In the case of incomes over $5000, the slope is 0.30. This steeper slope shows that for every extra dollar earned over $5000, the tax increases by 30 cents. The concept of slope in this situation helps students see the relationship between income changes and corresponding tax changes accurately, especially when visualized on a graph.