2. Lesson Menu Five-Minute Check (over Lesson 4–6) CCSS Then/Now New Vocabulary Key Concept: Inverse Relations Example 1: Inverse Relations Example 2: Graph Inverse Relations Key Concept: Finding Inverse Functions Example 3: Find Inverse Linear Functions Example 4: Real-World Example: Use an Inverse Function

3. Over Lesson 4–6 5-Minute Check 1 A.y = 4.3x + 15.2 B.y = 5.7x + 16.1 C.y = 2x + 13 D.y = x + 3 Which equation of a regression line best represents the table? Let x be the number of years since 2001.

4. Over Lesson 4–6 5-Minute Check 2 A.y = 25x + 13.2 B.y = 0.08x + 14.3 C.y = 0.16x + 14.61 D.y = 0.25x + 12 Which equation of a regression line best represents the table?

5. Over Lesson 4–6 5-Minute Check 3 A.y = 200.2x – 30 B.y = 190x – 20 C.y = 97x + 7 D.y = 54x – 18 Which equation of a regression line best represents the table?

6. Over Lesson 4–6 5-Minute Check 4 A.y = x + 70 B.y = 4x + 40 C.y = 5.5x + 92 D.y = 6.5x + 92 A line of best fit for a set of data has slope 6.5 and passes through the point at (–8, 40). What is the regression equation?

7. CCSS Content Standards A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. F.BF.4a Solve an equation of the form f (x ) = c for a simple function f that has an inverse and write an expression for the inverse. Mathematical Practices 6 Attend to precision. Common Core State Standards © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

8. Then/Now You represented relations as tables, graphs, and mappings. Find the inverse of a relation. Find the inverse of a linear function.

9. Vocabulary inverse relation – the set of ordered pairs obtained by exchanging the x-coordinates with the y-coordinates of each ordered pair in a relation inverse function – can generate ordered pairs of the inverse relation

10. Concept Notice that the domain of a relation becomes the range of its inverse, and the range of the relation becomes the domain of its inverse.

11. Find the inverse of each relation. 1){(4, -10), (7, -19), (-5, 17), (-3, 11)} To find the inverse, exchange the coordinates of the ordered pairs (4, -10) (-10, 4) (7, -19) (-19, 7) (-5, 17) (17, -5) (-3, 11) (11, -3) The inverse is {(-10, 4), (-19, 7), (17, -5), (11, -3)}

12. Write the coordinates as ordered pairs. Then exchange the coordinates of each pair. (-4, -13) (-13, -4) (-1, -8.5) (-8.5, -1) (5, 0.5) (0.5, 5) (9, 6.5) (6.5, 9) The inverse is {(-13, -4), (-8.5, -1), (0.5, 5), (6.5, 9)} x-459 y-13-8.50.56.5

13. Find the inverse of each relation. Write the ordered pairs {(-6, 8), (-15, 11), (9, 3), (0, 6)} Exchange the coordinates The inverse is {(8, -6), (11, -15), (3, 9), (6, 0)} x-6-1590 y81136

14. Example 1 Inverse Relations To find the inverse, exchange the coordinates of the ordered pairs. (–3, 26) → (26, –3) (6, –1) → (–1, 6) (2, 11) → (11, 2) (  1, 20) → (20,  1) A. Find the inverse of each relation. {(−3, 26), (2, 11), (6, −1), (−1, 20)} Answer: The inverse is {(26, –3), (11, 2), (–1, 6), (20, –1)}.

15. Example 1 Inverse Relations B. Find the inverse of each relation. Write the coordinates as ordered pairs. Then exchange the coordinates of each pair. (  4,  3) → (  3,  4) (–2, 0) → (0, –2) (1, 4.5) → (4.5, 1) (5, 10.5) → (10.5, 5) Answer: The inverse is {(3, 4), (4.5, 1), (0, –2), (10.5, 5)}.

16. Example 1 A.{(4, 8), (–6, 6), (3, 3), (0, –8)} B.{(8, 4), (6, –6), (3, 3), (–8, 0)} C.{(0, –8), (3, 3), (–6, 6), (4, 8)} D.{(–4, –8), (6, –6), (–3, –3), (0, 8)} Find the inverse of {(4, 8), (–6, 6), (3, 3), (0, –8)}.

17. Example 2 Graph Inverse Relations A. Graph the inverse of each relation.

18. Example 2 Answer: The graph of the relation passes through the points at (–2, 6), (2, 0), and (6, 6). To find points through which the graph of the inverse passes, exchange the coordinates of the ordered pairs. The graph of the inverse passes through the points at (6, –2), (0, 2), and (6, 6). Graph these points and then draw the line that passes through them. Graph Inverse Relations

19. Example 2 Graph Inverse Relations B. Graph the inverse of each relation.

20. Example 2 Answer: The graph of the relation passes through the points at (–2,– 6), (0, 4), (2, 0), (4, –4), and (6, –8). To find points through which the graph of the inverse passes, exchange the coordinates of the ordered pairs. The graph of the inverse passes through the points at (-6, -2), (4, 0), (0, 2), (–4, 4), and (–8, 6). Graph these points and then draw the line that passes through them. Graph Inverse Relations

21. Example 2 Graph the inverse of the relation.

22. Example 2 A.B. C.D.

23. Concept

24. Example 3 Find Inverse Linear Functions A. Find the inverse of the function f (x) = –3x + 27. Step 1 f(x)= –3x + 27Original equation y= –3x + 27Replace f(x) with y. Step 2 x = –3y + 27Interchange y and x. Step 3 x – 27 = –3ySubtract 27 from each side. x – 27 = y 3 3 Divide each side by –3.

25. Example 3 Simplify. Step 4 Answer: The inverse of f(x) = –3x + 27 is Find Inverse Linear Functions

26. Find the inverse of each function. f(x) = 4x – 8 Replace f(x) with y y = 4x – 8 Interchange y and x x = 4y – 8 Solve for y x = 4y – 8 +8

27. x + 8 = 4y 44 4 x/4 + 2 = y Replace y with f(x) -1 f(x) -1 = 1/4x + 2 The inverse of f)x) = 4x – 8 is f(x) -1 = 1/4x + 2 When graphed on the same coordinate plane f(x) -1 appears to be the reflection of f(x)

28. f(x) = 4x – 12 y = 4x – 12 x = 4y – 12 +12 + 12 x + 12 = 4y 4 4 4 y = 1/4x + 3 f(x) -1 = 1/4x + 3

29. Example 3 Step 1 Original equation Replace f(x) with y. Step 2 Interchange y and x. Step 3 Add 8 to each side. Find Inverse Linear Functions

30. Example 3 Answer: Step 4 Simplify. Find Inverse Linear Functions

31. f(x) = - ½ x + 11 y = - ½ x + 11 x = - ½ y + 11 -11 -11 x – 11 = - ½ y -2(x – 11) = - ½ y(-2) -2x + 22 = y f(x) -1 = -2x + 22

32. f(x) = 1/3x + 7 y = 1/3x + 7 x = 1/3y + 7 -7 -7 x – 7 = 1/3y 3(x – 7) = 1/3y(3) 3x – 21 = y f(x) -1 = 3x - 21

33. Example 3 Find the inverse of f(x) = 12 – 9x. A. B. C. D.

34. Find the inverse of each function 1)f(x) = 17 – 1/3x f(x) -1 = -3x + 51 2) f(x) = 12 – 6x f(x) -1 = -1/6x + 2 3) f(x) = -16 – 4/3x f(x) -1 = -3/4x – 12 4) f(x) = 5/11x + 10 f(x) -1 = 11/5x – 22 5) f(x) = -2/3x + ¼ f(x) -1 = -3/2x + 3/8

35. Example 4 Use an Inverse Function Step 1 f(x)= 2200 + 0.05xOriginal equation SALES Carter sells paper supplies and makes a base salary of $2200 each month. He also earns 5% commission on his total sales. His total earnings f(x) for a month in which he compiled x dollars in total sales is f(x) = 2200 + 0.05x. A. Find the inverse function. y = 2200 + 0.05xReplace f(x) with y. Step 2 x = 2200 + 0.05yInterchange y and x.

36. Example 4 Use an Inverse Function Step 3 x – 2200 = 0.05ySubtract 2200 from each side. Divide each side by 0.05. Step 4 Answer:

37. Example 4 Use an Inverse Function SALES Carter sells paper supplies and makes a base salary of $2200 each month. He also earns 5% commission on his total sales. His total earnings f(x) for a month in which he compiled x dollars in total sales is f(x) = 2200 + 0.05x. B. What do x and f –1 (x)represent in the context of the inverse function? Answer: x represents Carter’s total earnings for the month and f –1 (x) represents the total monthly sales by Carter for the company.

38. Example 4 Use an Inverse Function SALES Carter sells paper supplies and makes a base salary of $2200 each month. He also earns 5% commission on his total sales. His total earnings f(x) for a month in which he compiled x dollars in total sales is f(x) = 2200 + 0.05x. C. Find Carter’s total sales for last month if his earnings for that month were $3450. Carter earned $3450 for the month. To find Carter’s total sales for that month, find f –1 (3450).

39. Example 4 Use an Inverse Function f –1 (x)= 20x – 44,000Original equation f –1 (3450) = 20(3450) – 44,000total earnings = $3450 f –1 (3450) = 69,000 – 44,000Multiply. f –1 (3450) = 25,000Subtract. Answer: Carter had $25,000 in total sales for the month.

40. Randall is writing a report on Santiago, Chile, and he wants to include a brief climate analysis. He found a table of temperatures recorded in degrees Celsius. He knows that a formula for converting degrees Fahrenheit to degrees Celsius is C(x) = 5/9(x – 32). He will need to find the inverse function to convert from degrees Celsius to Fahrenheit.

41. (x) Average Temp. ( 0 C) MonthMinMax Jan1229 Mar927 May518 July315 Sept629 Nov926 Find the inverse function C -1 (x) C(x) = 5/9(x – 32)

42. y = 5/9(x – 32) x = 5/9(y – 32) x = 5/9y – 160/9 9/5x = 9/5(5/9y – 32) 9/5x = y – 32 +32 9/5x + 32 = y C -1 (x) = 9/5x + 32

43. What do x and C -1 (x) represent in the context of the inverse function? x represents the temperature in degrees Celsius and C -1 (x)n represents the temperature in degrees Fahrenheit.

44. Find the average temperatures for July in degrees Fahrenheit. The average min is 3 0 C and max is 15 0 C Find the average min and max for Fahrenheit C -1 (x) = 9/5x + 32 C -1 (3) = 9/5(3) + 32 = 37.4 C -1 (15) = 9/5(15) + 32 = 59 The average temp min is 37.4 0 F and max id 59 0 F for July

45. Peggy rents a car for the day. The total cost C(x) in dollars is given by C(x) = 19.99 + 0.3x, where x is the number of miles she drives. 1)Find the inverse function C -1 (x) = 1/3x – 19.99/3 2) What do x and C -1 (x) represent in the context of the inverse function? x is the total cost and C -1 (x) is the total number of miles driven 3) How many miles did Peggy drive if her total cost was $34.99? 50 miles

46. Example 4 A. B. C. D. REPAIRS Nikki’s car is getting repairs. The mechanic is charging her $40 to look at the car and $65 for each half-hour to fix the car. Her total cost f(x) for the repairs is f(x) = 40 + 65x. Find the inverse function and how long it took the mechanic to fix the car if Nikki was charged a total of $365.

47. End of the Lesson