1. Function Notation and Making Predictions Section 2.3

2. Lehmann, Intermediate Algebra, 3ed Section 2.3 Rather than use an equation, table, graph, or words to refer to a function: name it If f is the name use to represent y: We refer to “ ” as functional notation We substitute for y in the equation does not mean f times x Slide 2 Introduction Function Notation Reads, “f of x”

3. Lehmann, Intermediate Algebra, 3ed Section 2.3 Substitute 4 for x in the equation : With function notation, the input is and output is Substitute 4 for x into the equation : mean input leads to an output of Slide 3 Introduction Function Notation

4. Lehmann, Intermediate Algebra, 3ed Section 2.3 Notice that is of the form The number is the value of y when x is 4 To find, we say that we evaluate the function at Evaluate Slide 4 Introduction/Evaluating a Function Function Notation Example

5. Lehmann, Intermediate Algebra, 3ed Section 2.3 Could have used “g” to name the function Common symbols are f, g, and h. Slide 5 Evaluating a Function Function Notation Solution

6. Lehmann, Intermediate Algebra, 3ed Section 2.3 Find given Slide 6 Evaluating Functions Function Notation Example Solution

7. Lehmann, Intermediate Algebra, 3ed Section 2.3 Find given Slide 7 Evaluating Functions Function Notation Example Solution

8. Lehmann, Intermediate Algebra, 3ed Section 2.3 Find given Slide 8 Evaluating Functions Function Notation Example Solution

9. Lehmann, Intermediate Algebra, 3ed Section 2.3 Some input–output pairs of a function are shown in the table. Find and Slide 9 Using a Table to Find an Output and an Input Function Notation Example Solution The input of leads to So, Inputs leading to are So, for,

10. Lehmann, Intermediate Algebra, 3ed Section 2.3 Let. Find Slide 10 Using An Equation to Find an Output and an Input Function Notation Example Solution

11. Lehmann, Intermediate Algebra, 3ed Section 2.3 Substitute and solve for x: Slide 11 Using An Equation to Find an Output and an Input Function Notation Solution Continued

12. Lehmann, Intermediate Algebra, 3ed Section 2.3 Verify solution using graphing calculator Use table in Ask mode Slide 12 Using An Equation to Find an Output and an Input Function Notation Graphing Calculator

13. Lehmann, Intermediate Algebra, 3ed Blue arrow shows that the input leads to output of So, Section 2.3 A graph of a function is sketched. Find refers to when We want y when Slide 13 Using a Graph to Find the Values of x and f(x) Function Notation Example Solution

14. Lehmann, Intermediate Algebra, 3ed The line contains the point (0, 1) So, Section 2.3 A graph of a function is sketched. Find refers to when We want y when Slide 14 Using a Graph to Find the Values of x and f(x) Function Notation Example Solution

15. Lehmann, Intermediate Algebra, 3ed Red arrow shows output originates from the input So, Section 2.3 A graph of a function is sketched. Find x when Want is the value of x when Slide 15 Using a Graph to Find the Values of x and f(x) Function Notation Example Solution

16. Lehmann, Intermediate Algebra, 3ed Output originates from the input So, Section 2.3 A graph of a function is sketched. Find x when Want is the value of x when Slide 16 Using a Graph to Find the Values of x and f(x) Function Notation Example Solution

17. Lehmann, Intermediate Algebra, 3ed Section 2.3 The table shows the average salaries of professors at four- year colleges and universities. Let s be the professors’ average salary(in thousands Slide 17 Using an Equation of a Linear Model to Make Predictions Using Function Notation with Models Example of dollars) at t years since 1900. A possible model is 1. Verify that the above function is the model.

18. Lehmann, Intermediate Algebra, 3ed Section 2.3 Graph the model and the scattergram in the same viewing window Function seems to model the data well Slide 18 Using an Equation of a Linear Model to Make Predictions Using Function Notation with Models Solution Example Continued 2. Rewrite the equation with the function name f.

19. Lehmann, Intermediate Algebra, 3ed Section 2.3 t is the independent variable s is the dependent variable f is the function name, so we rewrite Substitute for s: Slide 19 Using an Equation of a Linear Model to Make Predictions Using Function Notation with Models Solution Example Continued 3. Predict the average salary in 2011.

20. Lehmann, Intermediate Algebra, 3ed Section 2.3 Represent the year 2011 by Substitute 111 for t into Slide 20 Using an Equation of a Linear Model to Make Predictions Using Function Notation with Models Solution Example Continued 4. Predict when the average salary will be $80,000.

21. Lehmann, Intermediate Algebra, 3ed Section 2.3 Represent average salary of $80,000 by Since, substitute 80 for and solve for t Slide 21 Using an Equation of a Linear Model to Make Predictions Using Function Notation with Models Solution

22. Lehmann, Intermediate Algebra, 3ed Section 2.3 According to model, average salary will be $80,000 in Using TRACE verify the predictions Slide 22 Using an Equation of a Linear Model to Make Predictions Using Function Notation with Models Graphing Calculator

23. Lehmann, Intermediate Algebra, 3ed Section 2.3 When making a prediction about the dependent variable of a linear model, substitute a chosen value for the independent variable in the model. Then solve for the dependent variable. When making a prediction about the independent variable of a linear model, substitute a chosen value for the dependent variable in the model. Then solve for the independent variable. Slide 23 Using an Equation of a Linear Model to make Predictions Using Function Notation with Models Summary

24. Lehmann, Intermediate Algebra, 3ed Section 2.3 To find a linear model and make estimates and predictions, 1.Create a scattergram of data to determine whether there is a nonvertical line that comes close to the data points. If so, choose two points (not necessarily data points) that you can use to find the equation of a linear model. 2.Find an equation of your model. Slide 24 Four-Step Modeling Process Using Function Notation with Models Process

25. Lehmann, Intermediate Algebra, 3ed Section 2.3 3.Verify your equation by checking that the graph of your model contains the two chosen points and comes close to all of the data points. 4.Use the equation of your model to make estimates, make predictions, and draw conclusions. Slide 25 Four-Step Modeling Process Using Function Notation with Models Process

26. Lehmann, Intermediate Algebra, 3ed Section 2.3 In an example from Section 2.2, we found the equation., where p is the percentage of Slide 26 Using Function Notation; Finding Intercepts Finding Intercepts Example American adults who smoke and t years since 1990. 1.Rewrite the equation with the function name g.

27. Lehmann, Intermediate Algebra, 3ed Section 2.3 To use the name g, substitute for p: 2.Find. What does the result mean in this function? Substitute 110 for t in the equation : Slide 27 Using Function Notation; Finding Intercepts Finding Intercepts Solution Example Continued Solution

28. Lehmann, Intermediate Algebra, 3ed Section 2.3 When t is 110, p is 16.2. According to the model, 16.2% of American adults will smoke in 2010. 3.Find the value of t when. What does is mean in this situation? Slide 28 Using Function Notation; Finding Intercepts Finding Intercepts Solution Continued Example Continued

29. Lehmann, Intermediate Algebra, 3ed Section 2.3 Substitute 30 for in the equation and solve for t When t is 110, p is 16.2. According to the model, 16.2% of American adults will smoke Slide 29 Using Function Notation; Finding Intercepts Finding Intercepts Solution Example Continued

30. Lehmann, Intermediate Algebra, 3ed Section 2.3 The model estimates that 30% of Americans smoked in Verify work on graphing calculator table Slide 30 Using Function Notation; Finding Intercepts Finding Intercepts Solution Continued Example Continued 4.Find the p-intercept of the model. What does it mean in this situation?

31. Lehmann, Intermediate Algebra, 3ed Section 2.3 Since the model is in slope- intercept form the p-intercept is (0, 74.50) The model estimates that 74.5% of American adults smoked in 1900 Research would show that this estimate is too high model breakdown has occurred Slide 31 Using Function Notation; Finding Intercepts Finding Intercepts Solution Example Continued 5.Find the t-intercept. What does it mean?

32. Lehmann, Intermediate Algebra, 3ed Section 2.3 To find the t-intercept, we substitute 0 for and solve for t: Slide 32 Using Function Notation; Finding Intercepts Finding Intercepts Solution

33. Lehmann, Intermediate Algebra, 3ed Section 2.3 The t-intercept is (140.57, 0) So, the model predicts that no Americans adults will smoke in Common sense suggest this probably won’t occur Use TRACE to verify the p- and i-intercepts. Slide 33 Using Function Notation; Finding Intercepts Finding Intercepts Solution Continued

34. Lehmann, Intermediate Algebra, 3ed Section 2.3 If a function of the form, where, is used to model a situation, then The p-intercept is (0, b). To find the coordinate of the t-intercept, substitute 0 for p in the model’s equation and solve for t. Slide 34 Intercepts of Models Finding Intercepts Property

35. Lehmann, Intermediate Algebra, 3ed Section 2.3 Sales of bagged salads increased approximately linearly from $0.9 billion in 1996 to $2.7 billion in 2004. Predict in which year the sales will be $4 billion. Let s be the sales (in billions of dollars) Let t be the years since 1990 We want an equation of the form Slide 35 Making a Prediction Using Data Described in Words to Make Predictions Example Solution

36. Lehmann, Intermediate Algebra, 3ed Section 2.3 First find the slope Substitute 0.23 for m: To find b we substitute 6 for t and 0.9 for s Slide 36 Making a Prediction Using Data Described in Words to Make Predictions Solution Continued

37. Lehmann, Intermediate Algebra, 3ed Section 2.3 Then substitute –0.48 for b: To predict when the sales will be $4 billion, we substitute 4 for s in the equation and solve for t: Slide 37 Making a Prediction Using Data Described in Words to Make Predictions Solution Continued

38. Lehmann, Intermediate Algebra, 3ed Section 2.3 The model predicts that sales will be $4 billion in Verify using a graphing calculator table Slide 38 Making a Prediction Using Data Described in Words to Make Predictions Solution Continued

39. Lehmann, Intermediate Algebra, 3ed Section 2.3 A store opens at 9 A. M. to 5 P. M., Mondays through Saturday. Let be an employee’s weekly income (in dollars) from working t hours each week at $10 per hour. 1.Find an equation of the model f. The employee’s weekly income (in dollars) is equal to the pay per hour times the number of hours worked per week: Slide 39 Finding the Domain and Range of a Function Domain and Range of a Function Example Solution

40. Lehmann, Intermediate Algebra, 3ed Section 2.3 2.Find the domain and range of the model f. To find domain and range we consider input-output Store is open 8 hours a day, 6 days a week, the employee can work up to 48 hours per week So, the domain is Since hours worked is between 0 and 48 hours, inclusively, the pay is between 0 and 48(10) Slide 40 Finding the Domain and Range of a Function Domain and Range of a Function Example Continued Solution

41. Lehmann, Intermediate Algebra, 3ed Section 2.3 Range is The figures illustrate inputs of 22, 35, and 48 being sent to the outputs 220, 350 and 480, respectively Label the t-axis that represents the domain and the part of the I-axis that represents the range Slide 41 Finding the Domain and Range of a Function Domain and Range of a Function Solution Continued