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Function Notation and Making Predictions

Section 2.3. Function Notation and Making Predictions. 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

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Function Notation and Making Predictions

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  1. Section 2.3 Function Notation and Making Predictions

  2. 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 Reads, “f of x” Section 2.3 Slide 2 Introduction Function Notation

  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 Section 2.3 Slide 3 Introduction Function Notation

  4. 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 Example Section 2.3 Slide 4 Introduction/Evaluating a Function Function Notation

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

  6. Example Find given Solution Section 2.3 Slide 6 Evaluating Functions Function Notation

  7. Example Find given Solution Section 2.3 Slide 7 Evaluating Functions Function Notation

  8. Example Find given Solution Section 2.3 Slide 8 Evaluating Functions Function Notation

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

  10. Example Let . Find Solution Section 2.3 Slide 10 Using An Equation to Find an Output and an Input Function Notation

  11. Solution Continued Substitute and solve for x: Section 2.3 Slide 11 Using An Equation to Find an Output and an Input Function Notation

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

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

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

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

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

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

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

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

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

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

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

  23. Summary 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. Section 2.3 Slide 23 Using an Equation of a Linear Model to make Predictions Using Function Notation with Models

  24. Process To find a linear model and make estimates and predictions, 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. Find an equation of your model. Section 2.3 Slide 24 Four-Step Modeling Process Using Function Notation with Models

  25. Process 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. Use the equation of your model to make estimates, make predictions, and draw conclusions. Section 2.3 Slide 25 Four-Step Modeling Process Using Function Notation with Models

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

  27. Solution 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 : Example Continued Solution Section 2.3 Slide 27 Using Function Notation; Finding Intercepts Finding Intercepts

  28. Solution Continued 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? Example Continued Section 2.3 Slide 28 Using Function Notation; Finding Intercepts Finding Intercepts

  29. Solution 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 Example Continued Section 2.3 Slide 29 Using Function Notation; Finding Intercepts Finding Intercepts

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

  31. Solution 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 Example Continued 5. Find the t-intercept. What does it mean? Section 2.3 Slide 31 Using Function Notation; Finding Intercepts Finding Intercepts

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

  33. Solution Continued 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. Section 2.3 Slide 33 Using Function Notation; Finding Intercepts Finding Intercepts

  34. Property 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. Section 2.3 Slide 34 Intercepts of Models Finding Intercepts

  35. Example 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 Solution Section 2.3 Slide 35 Making a Prediction Using Data Described in Words to Make Predictions

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

  37. Solution Continued 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: Section 2.3 Slide 37 Making a Prediction Using Data Described in Words to Make Predictions

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

  39. Example 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: Solution Section 2.3 Slide 39 Finding the Domain and Range of a Function Domain and Range of a Function

  40. Example Continued 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) Solution Section 2.3 Slide 40 Finding the Domain and Range of a Function Domain and Range of a Function

  41. Solution Continued 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 Section 2.3 Slide 41 Finding the Domain and Range of a Function Domain and Range of a Function

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