Problem 87 - Algebra and Trigonometry

Question

Income Tax In a certain country, the tax on incomes less than or equal to \(€ 20,000\) is 10\(\%\) . For incomes that are more than \(€ 20,000,\) the tax is \(€ 2000\) plus 20\(\%\) of the amount over \(€ 20,000\) . (a) Find a function \(f\) that gives the income tax on an income \(x .\) Express \(f\) as a piecewise defined function. (b) Find \(f^{-1} .\) What does \(f^{-1}\) represent? (c) How much income would require paying a tax of \(€ 10,000 ?\)

Step-by-Step Solution

Answer

(a) \( f(x) = \begin{cases} 0.10x, & x \leq 20000 \\ 0.20x - 2000, & x > 20000 \end{cases} \) (b) \( f^{-1}(y) = \begin{cases} 10y, & y \leq 2000 \\ 5y + 10000, & y > 2000 \end{cases} \), it represents the income needed for a given tax. (c) Income is \( €60,000 \).

Step 1: Define the Function

To find the tax function \( f \) for any given income \( x \), analyze the tax rules: - If \( x \leq €20,000 \), the tax is \( 10\% \) of \( x \). So, the tax is \( f(x) = 0.10x \).- If \( x > €20,000 \), the tax is \( €2000 \) plus \( 20\% \) of the excess amount over \( €20,000 \). The tax here is given by \( f(x) = 2000 + 0.20(x - 20000) = 0.20x - 2000 \). Combine these to form a piecewise function: \[ f(x) = \begin{cases} 0.10x, & x \leq 20000 \ 0.20x - 2000, & x > 20000 \end{cases} \]

Step 2: Solve for Inverse Function

Find the inverse \( f^{-1}(y) \) for the given function, where \( y = f(x) \).For \( y = 0.10x \), solve \( y = 0.10x \) implying \( x = \frac{y}{0.10} = 10y \).For \( y = 0.20x - 2000 \), solve \( y = 0.20x - 2000 \) implying \( 0.20x = y + 2000 \), hence \( x = \frac{y + 2000}{0.20} = 5y + 10000 \).Therefore, the inverse is: \[ f^{-1}(y) = \begin{cases} 10y, & y \leq 2000 \ 5y + 10000, & y > 2000 \end{cases} \]

Step 3: Interpretation of Inverse Function

The inverse function \( f^{-1} \) represents the income \( x \) needed to pay a given amount of tax \( y \). Essentially, it converts a tax amount back to the original income.

Step 4: Calculate Income for Specific Tax

Given the tax \( y = €10,000 \), use the second case of the inverse function to find income \( x \) since \( 10,000 > 2000 \).Apply \( x = 5y + 10000 \),\[ x = 5(10000) + 10000 = 50000 + 10000 = 60000 \]Thus, an income of \( €60,000 \) requires paying a tax of \( €10,000 \).

Key Concepts

Understanding the Income Tax Function

In this exercise, we encounter a classic example of a piecewise function. Piecewise functions help represent scenarios, like tax calculation, where different rules apply over different intervals.
By understanding the tax system:

• For incomes less than or equal to €20,000, the tax rate is 10%. Thus, the tax collected can be calculated with the formula: \( f(x) = 0.10x \).


• For incomes exceeding €20,000, a base tax of €2,000 applies, plus 20% of the income exceeding €20,000. This gives us the algebraic expression: \( f(x) = 2000 + 0.20(x - 20000) = 0.20x - 2000 \).

Combine these expressions into a piecewise function:\[f(x) = \begin{cases} 0.10x, & \text{if } x \leq 20000 \0.20x - 2000, & \text{if } x > 20000 \end{cases}\]
This form of representation is crucial in making the function understandable and applicable for calculating tax depending on different income brackets.

Decoding the Inverse Function

The inverse function, denoted as \( f^{-1}(x) \), takes output values back to their original input. In this case, it finds the income corresponding to a specified tax amount.

• For tax values up to €2,000, normal rule applies: \( y = 0.10x \), so the original income is \( x = \frac{y}{0.10} = 10y \).


• For tax values beyond €2,000, the calculation changes to: \( y = 0.20x - 2000 \). Solving gives us \( x = 5y + 10000 \).

Thus, the inverse function is given by the expression:\[f^{-1}(y) = \begin{cases} 10y, & \text{if } y \leq 2000 \5y + 10000, & \text{if } y > 2000 \end{cases}\]
This inverse function is incredibly useful when you need to determine which income will result in a specific tax amount.

Exploring Algebraic Expressions

Algebraic expressions form the foundation of understanding piecewise and inverse functions in calculus. In this context, an algebraic expression is a mathematical phrase combining numbers, variables, and operators to denote a specific calculation.
The piecewise function in the exercise uses algebraic expressions to define tax calculations for different income levels:

• In the first case, \( 0.10x \) is a simple algebraic expression representing 10% tax on incomes ≤ €20,000.


• For incomes above €20,000, the expression \( 0.20x - 2000 \) defines the tax, with €2,000 fixed and further calculated using 20% of the surplus.

Such expressions are powerful tools in various financial computations, providing clarity and precision. They enable both simplification in writing functions and provide insight into how different parameters affect outcomes.