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A graduated tax function is a tax structure where different portions of income are taxed at different rates. In this exercise, we consider a graduated tax function where the first \(\$30,000\) of income is taxed at \(10\%\), and any income above \(\$30,000\) is taxed at \(20\%\). This is a common approach in progressive tax systems aimed at ensuring higher earners contribute a larger share to the tax revenue. To calculate the tax:

- If an individual's income is \(\$30,000\) or less, they pay \(10\%\) of their income as tax.
- If the income exceeds \(\$30,000\), the first \(\$30,000\) is taxed at \(10\%\) and the amount over \(\$30,000\) is taxed at \(20\%\).
Let's see an example:

For an income of \(\$40,000\):
- The first \(\$30,000\) is taxed at \(10\%\), yielding \(0.10 \times 30,000 = \$3,000\).
- The next \(\$10,000\) is taxed at \(20\%\), yielding \(0.20 \times 10,000 = \$2,000\).
Hence, the total tax is \(\$3,000 + \$2,000 = \$5,000\).

A flat tax function is simpler than a graduated tax function. Here, the tax rate is constant regardless of income level. In this exercise, the flat tax rate is set at \(15\%\). This means every individual pays the same percentage of their income in taxes, which simplifies calculations but does not account for the ability to pay in the same way as progressive taxation.

To calculate the tax for a given income, you simply multiply the income by the tax rate. Here are a few examples:
- For an income of \(\$10,000\), the tax is \(0.15 \times 10,000 = \$1,500\).
- For an income of \(\$30,000\), the tax is \(0.15 \times 30,000 = \$4,500\).
This structure is easy to compute but is often seen as less equitable than graduated taxes since all income levels are taxed at the same rate.

To visually compare the graduated and flat tax functions, you can graph them on the same grid. The x-axis represents income, while the y-axis represents the tax amount.

For the graduated tax function:
- The tax increases at a steeper rate after \(\$30,000\).
- The graph is a piecewise function with a slope change at \(\$30,000\).

For the flat tax function:
- The tax amount increases linearly with income at a constant rate.

To graph these:
1. For the graduated tax, plot points up to \(\$30,000\) at \(10\%\) and after \(\$30,000\) at \(20\%\).
2. For the flat tax, plot points along a straight line at \(15\%\) of income.

The point where these graphs intersect represents the income level where the tax from both structures is the same. Identifying this point is crucial for understanding at which income level the tax burden shifts between the two structures.

Income tax calculations can vary significantly depending on the tax structure. Understanding how to compute taxes under different systems helps in better financial planning.

Let’s compare both tax functions for specific incomes as illustrated in the exercise:

- For \(\$10,000\):
- Graduated tax: \(0.10 \times 10,000 = \$1,000\).
- Flat tax: \(0.15 \times 10,000 = \$1,500\).

- For \(\$30,000\):
- Graduated tax: \(0.10 \times 30,000 = \$3,000\).
- Flat tax: \(0.15 \times 30,000 = \$4,500\).

- For \(\$50,000\):
- Graduated tax: \(0.10 \times 30,000 + 0.20 \times 20,000 = \$7,000\).
- Flat tax: \(0.15 \times 50,000 = \$7,500\).

These calculations show the distinct tax liabilities individuals face under different tax systems. Graduated taxes often result in lower total taxes for lower income brackets, while flat taxes apply uniformly across all income levels.