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1.2. Functions and Their Properties. Quick Review. Quick Review Solutions. Homework, Page 81. Match the numerical model to the corresponding graphical model and algebraic model. 3. a. p. Homework, Page 81.

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  1. 1.2 Functions and Their Properties

  2. Quick Review

  3. Quick Review Solutions

  4. Homework, Page 81 Match the numerical model to the corresponding graphical model and algebraic model. 3. a. p.

  5. Homework, Page 81 Match the numerical model to the corresponding graphical model and algebraic model. 7. g. s.

  6. Homework, Page 81 Refer to the table showing the percentage of the male and female populations employed in the U.S. civilian work force for selected years. 11.

  7. Homework, Page 81 11. According to the numerical model, what has been the trend of females entering the workforce since 1954? The percentage to the female population in the workforce has steadily risen since 1954.

  8. Homework, Page 81 11. In what five year period did the percentage of women who were employed change the most? The percentage to the female population in the workforce changed the most from 1974 to 1979.

  9. Homework, Page 81 15. Model the data algebraically with linear equations. Write one equation for women and one for men. Use the 1954 and 1999 data to compute slopes.

  10. Homework, Page 81 15.

  11. Homework, Page 81 19. Enter the data from the total column in list L1 and the female column in L2.

  12. Homework, Page 81 19. a. Enter the command: b. At the top of L2, enter:

  13. Homework, Page 81 23. A physics student obtains the data in the table from rolling a ball down an inclined plane. Find an algebraic model that fits the data.

  14. Homework, Page 81 27. The owners attempted to halt uncontrolled spending by proposing a salary cap which prompted a players’ strike in 1994. The strike caused the 1995 season to be shortened. Identify where the 1995 salaries appear in the graph and explain how you can spot them. The 1995 salaries appear at year 15. this is apparent, since the players would earn less for fewer games and the salary cap had not yet been lifted, thus a dip in average pay.

  15. Homework, Page 81 Solve the equation algebraically and graphically. 31.

  16. Homework, Page 81 Solve the equation algebraically and graphically. 35.

  17. Homework, Page 81 Solve the equation graphically by converting to the form f(x) =0 and finding the x-intercepts. 39.

  18. Homework, Page 81 Solve the equation graphically by converting to the form f(x) =0 and finding the x-intercepts. 43.

  19. Homework, Page 81 47. Swan Auto charges $32/day plus $0.18/mile. a. Elaine rented a car for one day and drove 83 miles. How much did she pay? b. Ramon paid $69.80 for one day. How far did he drive?

  20. Homework, Page 81 51. Solving graphically, find all real solutions for the equations. a.

  21. Homework, Page 81 51. Solving graphically, find all real solutions for the equations. b.

  22. Homework, Page 81 59. Select which equation fits the data. A. B. C. D. E.

  23. Homework, Page 81 63. The chart shows the number of cell phones and the average monthly bill for their use.

  24. Homework, Page 81 63. a. Graph a scatter plot of the number of subscribers and average monthly bills as a function of time, letting t = number of years after 1990.

  25. Homework, Page 81 63. b. One plot suggests a linear model. Find the linear model.

  26. Homework, Page 81 63. c. Superimpose the linear model on the scatter plot. Does the fit appear good? The fit of the model appears to be good.

  27. Homework, Page 81 63. d. Why does a linear model seem inappropriate for the other scatter plot? What function might fit better? The scatter plot for average monthly bills seems to have more curve to it and might be better fit by a quadratic, logarithmic, or square or cube root function.

  28. Homework, Page 81 63. e. In 195, there were 33.8 million subscribers with an average monthly bill of $51.00. Add these points to the scatter plot.

  29. Homework, Page 81 63. f. The 1995 data do not seem to fit the model. Give a possible explanation. The initial buyers of cell phones were commercial interests that were willing to pay the higher rates. As competition brought rates down, more individuals began purchasing cell phones leading to the observed growth in numbers.

  30. What you’ll learn about • Function Definition and Notation • Domain and Range • Continuity • Increasing and Decreasing Functions • Boundedness … and why Functions and graphs form the basis for understanding the mathematics and applications you will see both in your work place and in coursework in college.

  31. Leading Questions (True or False) A continuous function is one with a unique y-value for each x-value. Functions are bound only if their values are constrained between two y-values. Extrema refer to minimum or maximum values of a function. Functions are bounded if their values are constrained by an upper limit and a lower limit.

  32. Function, Domain, and Range A function from a set D to a set R is a rule that assigns to every element in D a unique element in R. The set D of all input values is the domain of the function, and the set R of all output values is the range of the function.

  33. Mapping

  34. Example Seeing a Function Graphically

  35. Vertical Line Test A graph (set of points (x, y)) in the xy-plane defines y as a function of x if and only if no vertical line intersects the graph in more than one point. The variable x is referred to as the independent variable and y is referred to as the dependent variable.

  36. Agreement Unless we are dealing with a model that necessitates a restricted domain, we will assume that the domain of a function defined by an algebraic expression is the same as the domain of the algebraic expression, the implied domain. For models, we will use a domain that fits the situation, the relevant domain.

  37. Example Finding the Domain of a Function

  38. Example Finding the Range of a Function

  39. Continuity

  40. Example Identifying Points of Discontinuity Which of the following figures shows functions that are discontinuous at x = 2?

  41. Continuity • A function is continuous if the point (x, f(x)) approaches the point (a, f(a)) ever more closely as x approaches a ever more closely from either direction. • Algebraically, we say • A function that is not continuous at a point is said to be discontinuous.

  42. Increasing and Decreasing Functions

  43. Increasing, Decreasing, and Constant Function on an Interval A function f is increasing on an interval if, for any two points in the interval, a positive change in x results in a positive change in f(x). A function f is decreasing on an interval if, for any two points in the interval, a positive change in x results in a negative change in f(x). A function f is constant on an interval if, for any two points in the interval, a positive change in x results in zero change in f(x).

  44. Example Analyzing a Function for Increasing-Decreasing Behavior

  45. Lower Bound, Upper Bound and Bounded A function f is bounded below if there is some number b that is less than or equal to every number in the range of f. Any such number b is called a lower bound of f. A function f is bounded above if there is some number B that is greater than or equal to every number in the range of f. Any such number B is called a upper bound of f. A function f is bounded if it is bounded both above and below.

  46. Trailing Questions (True or False)

  47. Homework • Review section 1.2 (pages 86 – 95) • Page 102, Exercises: 1 – 37 (EOO), 71 – 77 (Odd)

  48. What you’ll learn about • Local and Absolute Extrema • Symmetry • Asymptotes • End Behavior … and why Functions and graphs form the basis for understanding The mathematics and applications you will see both in your work place and in coursework in college.

  49. Local and Absolute Extrema A local maximum of a function f is a value f(c) that is greater than or equal to all range values of f on some open interval containing c. If f(c) is greater than or equal to all range values of f, then f(c) is the maximum (or absolute maximum) value of f. A local minimum of a function f is a value f(c) that is less than or equal to all range values of f on some open interval containing c. If f(c) is less than or equal to all range values of f, then f(c) is the minimum (or absolute minimum) value of f. Local extrema are also called relative extrema.

  50. Example Identifying Local Extrema

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