1. The problem is to understand the Federal Tax Rates for Individuals based on their taxable income brackets. 2. The tax rate is applied progressively to income within each bracket, meaning income is taxed at different rates as it moves through brackets. 3. The tax brackets and rates are: - 10% for income $0$ to $9875$ - 12% for income $9876$ to $40125$ - 22% for income $40126$ to $85525$ - 24% for income $85526$ to $163300$ - 32% for income $163301$ to $207350$ - 35% for income $207351$ to $518400$ - 37% for income $518401$ or more 4. To calculate the total tax for an income $I$, sum the tax for each bracket up to $I$: $$\text{Tax} = \sum_{k} \text{rate}_k \times \text{(income in bracket } k)$$ 5. For example, if $I = 50000$, tax is: - 10% on first $9875$ - 12% on next $40125 - 9875 = 30250$ - 22% on remaining $50000 - 40125 = 9875$ 6. The tax function is a piecewise linear function with increasing slopes at bracket thresholds, forming a stepwise increasing graph. 7. The graph of tax rate vs taxable income bracket is a step function increasing at the bracket boundaries.