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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1.4 Building Functions from Functions

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 2 Quick Review

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 3 Quick Review Solutions

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 4 Homework, Page 113 Each graph is a slight variation of one of the 12 basic functions. Match the graph to one of the 12 functions. 1.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 5 Homework, Page 113 Each graph is a slight variation of one of the 12 basic functions. Match the graph to one of the 12 functions. 5.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 6 Homework, Page 113 Each graph is a slight variation of one of the 12 basic functions. Match the graph to one of the 12 functions. 9.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 7 Homework, Page 113 Identify which of the Exercise 1 – 12 displays fit the description given. 13.The function whose domain excludes zero. 8.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 8 Homework, Page 113 Identify which of the Exercise 1 – 12 displays fit the description given. 17.The six functions that are bounded below. 2. 4.6. 10. 11. 12.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 9 Homework, Page 113 Identify which of the 12 basic functions fit the description given. 21.The three functions that are decreasing on the interval (-∞, 0). The squaring function, the reciprocal function, and the absolute value function are all decreasing on the interval (-∞, 0).

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 10 Homework, Page 113 Identify which of the 12 basic functions fit the description given. 25.The four functions that do not have end behavior. The four functions that do not have end behavior are the reciprocal function, the sine function, the cosine function, and the logistic function.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 11 Homework, Page 113 Use your graphing calculator to produce a graph of the function. Determine the domain and range of the function. 29.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 12 Homework, Page 113 Use your graphing calculator to produce a graph of the function. Determine the domain and range of the function. 33.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 13 Homework, Page 113 In Exercises 37 and 41, graph the equations, then answer the following questions: a. On what interval, if any, is the function increasing? Decreasing? b. Is the function odd, even, or neither. c. Give the function’s extrema, if any. d. How does the graph relate to a graph of one of the 12 basic functions?

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 14 Homework, Page 113 37. a. The function is increasing on its domain, (-∞, ∞). It is decreasing nowhere. b. The function is neither odd nor even. c. The function has no extrema. d. The graph has three times the magnitude of the logistic function’s graph.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 15 Homework, Page 113 41. a. The function is increasing on [2, ∞) and decreasing on (-∞, 2]. b. The function is neither odd nor even. c. The function has an absolute minimum at x = 2. d. The graph is the absolute value function’s graph translated two units to the right.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 16 Homework, Page 113 Graph the piece-wise defined function. Give any points of discontinuity. 45. No discontinuities.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 17 Homework, Page 113 Graph the piece-wise defined function. Give any points of discontinuity. 49. No discontinuities.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 18 Homework, Page 113 53. The function is one of the 12 basic functions, written another way. a. Graph the function and identify which basic function it is. The absolute value function

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 19 Homework, Page 113 53. b. Explain algebraically why the two functions are equal.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 20 Homework, Page 113 59. The logistic function has two horizontal asymptotes. Justify your answer. True, the left and right end behavior models are different, so there are two asymptotes.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 21 Homework, Page 113 61. Which function is bounded both above and below? a.b. c.d. e.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 22 Homework, Page 113 63.Which function is increasing on the interval (–∞, ∞)? A. B. C. D. E.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 23 What you’ll learn about Combining Functions Algebraically Composition of Functions Relations and Implicitly Defined Functions … and why Most of the functions that you will encounter in calculus and in real life can be created by combining or modifying other functions.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 24 Sum, Difference, Product, and Quotient

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 25 Example Defining New Functions Algebraically

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 26 Composition of Functions

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 27 Composition of Functions

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 28 Example Composing Functions

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 29 Example Decomposing Functions

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 30 Example Using Implicitly Defined Functions

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 31 Homework Homework Assignment #5 Read Section 1.5 Page 124, Exercises: 1 – 53 (EOO)

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1.5 Parametric Relations and Inverses

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 33 Quick Review

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 34 Quick Review Solutions

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 35 What you’ll learn about Defining Relations Parametrically Inverse Relations Inverse Functions … and why Some functions and graphs can best be defined parametrically, while some others can be best understood as inverses of functions we already know.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 36 Parametric Equations A parametric equation is one that defines the two elements of an ordered pair in terms of a third variable, called the parameter. A pair of parametric equations may define either a function or a relation. Remember, a function is a relation that passes the vertical line test.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 37 Example Defining a Function Parametrically

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 38 Example Defining a Function Parametrically

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 39 Inverse Relation The ordered pair (a,b) is in a relation if and only if the pair (b,a) is in the inverse relation.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 40 Horizontal Line Test The inverse of a relation is a function if and only if the relation is a function and any horizontal line intersects the graph of the original relation at no more than one point. A function that passes this horizontal line test is called a one-to-one function.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 41 Inverse Function

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 42 Example Finding an Inverse Function Algebraically

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 43 The Inverse Reflection Principle The points (a,b) and (b,a) in the coordinate plane are symmetric with respect to the line y = x. The points (a,b) and (b,a) are reflections of each other across the line y = x.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 44 The Inverse Composition Rule

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 45 Example Verifying Inverse Functions

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 46 How to Find an Inverse Function Algebraically

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