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Understanding Piecewise-Defined Functions through Tax and Paint Examples

This section delves into piecewise-defined functions using practical examples, including U.S. income tax rates and paint coverage calculations. It outlines the income tax structure based on different brackets, illustrating how to compute taxes for various incomes, and encourages graphing income tax paid as a function of taxable income. Additionally, the section discusses calculating paint needed for a given area and offers insights into the absolute value function, including its domain and range. Real-world applications emphasize understanding functions in varied contexts.

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Understanding Piecewise-Defined Functions through Tax and Paint Examples

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  1. Math 170 Functions, Data, and Models 09 Piecewise-Defined Functions Section 2.3 Comments on Lab 02.

  2. Basic Concept • A function can be defined piecewise.

  3. Tax Example • The U.S. Income Tax rates for the first three tax brackets are • 10% on taxable income from $0 to $8,700, • 15% on taxable income over $8,700 to $35,350 • 25% on taxable income over $35,350 to $85,650 • Find the income tax paid on taxable incomes of $1,000, $10,000, and $50,000. • Draw a graph of income tax paid as a function of taxable income. • Find a formula for income tax paid as a function of taxable income. • How can it be that 47% of adult pay no federal income tax?

  4. Paint Example • Consider , where is the number of gallon cans of paint to be bought to cover a surface of square meters. Suppose 1 gallon of paint covers 9 square meters. • What is , , and • Draw a graph of on the domain . • Find a formula for on the domain .

  5. Graph Example • Find a formula for the given graph.

  6. Formula Examples • Suppose . Graph on a suitable domain. • Suppose g. Is a function? • Suppose h. Is a function?

  7. Absolute Value Example • The absolute value of , denoted , is the distance is from the origin. • Find , , and . • Graph the absolute value function. • Find a formula for the absolute value function. • What is the domain and range of the absolute value function?

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