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Understanding Piecewise-Defined Functions: Concepts and Examples

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Dive into the concept of piecewise-defined functions with this comprehensive quiz and example set. Learn how these functions are structured using different rules over varied subsets of their domain. Explore typical examples like f(x) = |x| and methods for graphing them accurately. Test your knowledge with questions regarding domain, range, and specific function evaluations. This resource includes practice problems and thorough explanations to help you master the topic. Perfect for students preparing for their exams in mathematics!

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Understanding Piecewise-Defined Functions: Concepts and Examples

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  1. 2.5 Piecewise- defined Functions

  2. Quiz • Have you taken your Exam 1 yet?

  3. Piecewise-Defined Function • example y 5 4 1 x 1 5 -2 f(x) = x2 f(x) = x2 if -2 ≤ x ≤ 1 f(x) = x if 1 < x ≤ 5 f(x) = x2 if -2 ≤ x ≤ 1 x if 1 < x ≤ 5 f(x) = x

  4. Piecewise-defined Function • Definition: a Piecewise-defined Function is a function defined by different rules over different subsets of its domain • Typical example: f(x) = |x| we can rewrite f(x) = |x| into piecewise-defined form as: f(x) = x if x ≥ 0 -x if x < 0

  5. Graph a piecewise-defined Function • Example: f(x) = x + 3 for -3 ≤ x < -1 Notice: When meeting with ‘<’ or ‘>’, use ‘ 。’to mark the end point . Other cases, use ‘ . ’. 5 for -1 ≤ x ≤ 1 √ x for 1 < x < 9 1, What is the domain? 2, What is the range? 3, Find f(0) 4, Find f(-5) 5, Find f(-1)

  6. Graph of the Piecewise-defined Function • Sketch the graph of the piecewise defined function: 4 for x ≤ 0 f(x) = - x2 for 0 < x ≤ 2 2x - 6 for x > 2

  7. Find The Formula For a Piecewise-defined Function • Example: y x f(x) = -x2 +3 if x ≤ 0 (1/3)x-1 if x > 0

  8. Homework • PG. 132: 6-24(M3), 33, 36, 37, 52 • KEY: 15, 36, 52 • Reading: 2.6 Combinations

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