Problem 82 - 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) $2500. Graph is piecewise with a kink at $6000.

Step 1: Calculate Tax for $3000

Since \(3000 is less than \)6000, we use the formula for incomes below \(6000: \( f(x) = 0.15x \). Substitute \( x = 3000 \) into the formula: \( f(3000) = 0.15 \times 3000 = 450 \). Thus, the tax for \)3000 is $450.

Step 2: Calculate Tax for $6000

For \(6000, we switch to the second formula since \)6000 falls at the boundary: \( f(x) = 900 + 0.40(x - 6000) \). Substitute \( x = 6000 \): \( f(6000) = 900 + 0.40 \times (6000 - 6000) = 900 \). The tax on an income of \(6000 is \)900.

Step 3: Calculate Tax for $10,000

An income of \(10,000 is greater than \)6000, so we use the formula for incomes above \(6000: \( f(x) = 900 + 0.40(x - 6000) \). Substitute \( x = 10000 \): \( f(10000) = 900 + 0.40 \times (10000 - 6000) = 900 + 0.40 \times 4000 = 900 + 1600 = 2500 \). Therefore, the tax for \)10,000 is $2500.

Step 4: Graph the Function

To graph the function, plot two separate line segments. For \( 0 \leq x < 6000 \), the line is \( y = 0.15x \). For \( x \geq 6000 \), the line is \( y = 900 + 0.40(x - 6000) \). The first segment starts at the origin \((0,0)\) and ends at \((6000, 900)\). The second segment starts at \((6000, 900)\) and continues with a slope of 0.40. The graph shows a continuous function changing slope at \( x = 6000 \).

Key Concepts

Income Tax Calculation

Calculating income tax using a piecewise function involves understanding which part of the function applies to your specific income level. Let's break it down with our example function. The function changes rules based on whether your income is below or equal to/exceeds \(6000.

• If your income is below \)6000, the tax calculation is straightforward: multiply your income by 15%. This is shown in the formula as \( f(x) = 0.15x \).
• For example, if you earn \\(3000, your tax is \( 0.15 \times 3000 = 450 \).
• For incomes of \)6000 or more, the tax includes a base amount plus 40% of the income over \(6000. This is expressed as \( f(x) = 900 + 0.40(x - 6000) \).
• Using this formula, a \)10,000 income results in a tax of \( 900 + 0.40 \times (10000 - 6000) = 2500 \). Each calculation adds clarity to how taxes are compounded at higher income levels.

Graphing Piecewise Functions

Graphing a piecewise function is a key skill that allows a clear visual representation of how a function behaves across different intervals. For the given income tax example, visualizing helps understand at which point the switch between formulas occurs.

• Start by graphing each part of the piecewise function separately. For the interval where \( 0 \leq x < 6000 \), the line's equation is \( y = 0.15x \). It originates from point \((0,0)\) up to \((6000, 900)\).
• For \( x \geq 6000 \), the function changes to \( y = 900 + 0.40(x - 6000) \), which plots a line continuing from \((6000, 900)\) with a slope of 0.40.
• The graph showcases the piecewise nature clearly by showing two segments, each representing a different tax calculation rule based on income brackets.

Continuity of Functions

Continuity in piecewise functions examines whether or not the function experiences any breaks or gaps as it transitions from one piece to the next. In our income tax function example, it appears that the function is continuous.

• A function is continuous if you can draw its graph without lifting your pen. At \( x = 6000 \), the transition point must align perfectly between the two rules.
• Using the piecewise formulas, both conditions at this switch point were computed to equal the same tax value of \( 900 \). Hence, the function is continuous.
• This continuity means no sudden jumps in tax rate occur around the transition point, which would be unfair or unexpected in real-world applications such as tax rates.

Understanding continuity helps anticipate adjustments at function boundaries and ensures smooth transitions in calculated outcomes.