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Homework, Page 124

Homework, Page 124. Find the formulas for f + g, f – g, and fg . Give the domain of each. 1. . Homework, Page 124. Find the formulas for f / g, and g / f . Give the domain of each. 5. . Homework, Page 124.

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Homework, Page 124

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  1. Homework, Page 124 Find the formulas for f + g, f – g, and fg. Give the domain of each. 1.

  2. Homework, Page 124 Find the formulas for f / g, and g / f. Give the domain of each. 5.

  3. Homework, Page 124 9. and are shown in a[0. 5] by [0, 5] viewing window. Sketch the graph of the sum

  4. Homework, Page 124 Find and 13.

  5. Homework, Page 124 Find and and find the domain of each. 17.

  6. Homework, Page 124 Find and and find the domain of each. 21.

  7. Homework, Page 124 Find and so that the function can be described as . 25.

  8. Homework, Page 124 Find and so that the function can be described as . 29.

  9. Homework, Page 124 33. A satellite camera takes a rectangular-shaped picture. The smallest region that can be photographed is a 5-km by 7-km rectangle. As the camera zooms out, the length and width of the rectangle increase at the rate of 2 km/sec. How long does it take for the area A to be at least five times the original size?

  10. Homework, Page 124 33.

  11. Homework, Page 124 Find two functions defined implicitly by the given relations. 37.

  12. Homework, Page 124 Find two functions implicitly defined by the given relations. 41.

  13. Homework, Page 124 45. True or False. The domain of the quotient function consists of all numbers that belong to both the domain of f and the domain of g. Justify your answer. False. The domain of the quotient function consists of all numbers that belong to both the domain of f and the domain ofg, less those that result in.

  14. Homework, Page 124 49. If , then a. b. c. d. e.

  15. Homework, Page 124 53. An identity for a function operation is a function that combines with a given function f to return the same function f. Find the identity functions for the following operations. a. Find a function g such that b. Find a function g such that c. Find a function g such that

  16. Homework, Page 124 53. a. Find a function g such that b. Find a function g such that

  17. Homework, Page 124 53. c. Find a function g such that

  18. 1.5 Parametric Relations and Inverses

  19. Quick Review

  20. Quick Review Solutions

  21. 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.

  22. 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.

  23. Example Defining a Function Parametrically

  24. Example Defining a Function Parametrically

  25. Inverse Relation The ordered pair (a,b) is in a relation if, and only if, the pair (b,a) is in the inverse relation.

  26. Horizontal Line Test The inverse of a relation is a function if, and only if, 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.

  27. Inverse Function

  28. Example Finding an Inverse Function Algebraically

  29. 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.

  30. The Inverse Composition Rule

  31. Example Verifying Inverse Functions

  32. How to Find an Inverse Function Algebraically

  33. Example Finding Inverse Functions

  34. Homework • Homework Assignment #5 • Review Section 1.6 • Page 135, Exercises: 1 – 49 (EOO), skip 45 • Quiz next time

  35. 1.6 Graphical Transformations

  36. Quick Review

  37. Quick Review Solutions

  38. What you’ll learn about • Transformations • Vertical and Horizontal Translations • Reflections Across Axes • Vertical and Horizontal Stretches and Shrinks • Combining Transformations … and why Studying transformations will help you to understand the relationships between graphs that have similarities but are not the same.

  39. Transformations Rigid transformation – an action that changes a graph in a predictable manner. The shape and the size of the graph remain unchanged, but its position changes horizontally, vertically or diagonally. Non-rigid transformation – generally a distortion of the shape of a graph, including horizontal or vertical stretches and shrinks

  40. Translations Let c be a positive real number. Then the following transformations result in translations of the graph of y=f(x). Horizontal Translations y=f(x-c) a translation to the right by c units y=f(x+c) a translation to the left by c units Vertical Translations y=f(x)+c a translation up by c units y=f(x)-c a translation down by c units

  41. Example Vertical Translations

  42. Example Finding Equations for Translations

  43. Reflections The following transformations result in reflections of the graph of y = f(x): Across the x-axis y = – f(x) Across the y-axis y = f(– x)

  44. Graphing Absolute Value Compositions Given the graph of y = f(x), the graph y = |f(x)| can be obtained by reflecting the portion of the graph below the x-axis across the x-axis, leaving the portion above the x-axis unchanged; the graph of y = f(|x|) can be obtained by replacing the portion of the graph to the left of the y-axis by a reflection of the portion to the right of the y-axis across the y-axis, leaving the portion to the right of the y-axis unchanged. (The result will show even symmetry.)

  45. Compositions With Absolute Value Match the compositions of y = f(x) with the graphs. 1. 2. 3. 4. a. b. c. d. e. f.

  46. Stretches and Shrinks Let c be a positive real number. Then the following transformations result in stretches or shrinks of the graph y = f(x). Horizontal Stretches or Shrinks Vertical Stretches and Shrinks

  47. Example Finding Equations for Stretches and Shrinks

  48. Example Combining Transformations in Order

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