1. Splash Screen

2. Lesson Menu Five-Minute Check (over Lesson 6–1) CCSS Then/Now New Vocabulary Key Concept: Inverse Relations Example 1:Find an Inverse Relation Key Concept: Property of Inverses Example 2:Find and Graph an Inverse Key Concept: Inverse Functions Example 3:Verify that Two Functions are Inverses

3. Over Lesson 6–1 5-Minute Check 1 A.(f – g)(x) = 2x 2 + 3x + 1 B.(f – g)(x) = 2x 2 + 3x + 3 C.(f – g)(x) = –x 2 – x – 1 D.(f – g)(x) = –2x 2 + 3x + 3 Using f(x) = 3x + 2 and g(x) = 2x 2 – 1, find (f – g)(x), if it exists.

4. Over Lesson 6–1 5-Minute Check 2 A.(f ● g)(x) = 6x 3 + 4x 2 – 3x – 2 B.(f ● g)(x) = 6x 2 + 4x – 2 C.(f ● g)(x) = 5x 2 + 4x – 3 D.(f ● g)(x) = x 2 + 6x + 1 Using f(x) = 3x + 2 and g(x) = 2x 2 – 1, find (f ● g)(x), if it exists.

5. Over Lesson 6–1 5-Minute Check 3 A.[f ○ g](x) = 12x 2 + 6x + 1 B.[f ○ g](x) = 6x 2 + 4x – 2 C.[f ○ g](x) = 6x 2 – 1 D.[f ○ g](x) = 3x 2 – 2 Using f(x) = 3x + 2 and g(x) = 2x 2 – 1, find [f ○ g](x), if it exists.

6. Over Lesson 6–1 5-Minute Check 4 A.[g ○ f](x) = 18x 3 + 24x 2 + 7x + 1 B.[g ○ f](x) = 18x 2 + 24x + 7 C.[g ○ f](x) = 6x 2 + 24x + 7 D.[g ○ f](x) = 6x 3 + 4x 2 – 3x – 2 Using f(x) = 3x + 2 and g(x) = 2x 2 – 1, find [g ○ f](x), if it exists.

7. Over Lesson 6–1 5-Minute Check 5 A.[g ○ f](x) = 0.75x + 15 B.[g ○ f](x) = 0.75(x + 20) + 15 C.[g ○ f](x) = x + 20(0.75x + 15) D.[g ○ f](x) = 1.75x + 35 To obtain a retail price, a dress shop adds $20 to the wholesale cost x of every dress. When the shop has a sale, every dress is sold for 75% of the retail price. If f(x) = x + 20 and g(x) = 0.75x, find [g ○ f](x) to describe this situation.

8. Over Lesson 6–1 5-Minute Check 6 A.f(1) B.g(1) C.(g ○ f)(1) D.f(0) Let f(x) = x – 3 and g(x) = x 2. Which of the following is equivalent to (f ○ g)(1)?

9. CCSS Content Standards F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. F.BF.4.a Find inverse functions. - 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 7 Look for and make use of structure. 8 Look for and express regularity in repeated reasoning.

10. Then/Now You transformed and solved equations for a specific variable. Find the inverse of a function or relation. Determine whether two functions or relations are inverses.

11. Vocabulary inverse relation inverse function

12. Concept

13. Example 1 Find an Inverse Relation GEOMETRY The ordered pairs of the relation {(1, 3), (6, 3), (6, 0), (1, 0)} are the coordinates of the vertices of a rectangle. Find the inverse of this relation. Describe the graph of the inverse. To find the inverse of this relation, reverse the coordinates of the ordered pairs. The inverse of the relation is {(3, 1), (3, 6), (0, 6), (0, 1)}.

14. Example 1 Find an Inverse Relation Answer: Plotting the points shows that the ordered pairs also describe the vertices of a rectangle. Notice that the graph of the relation and the inverse are reflections over the graph of y = x.

15. Example 1 A.cannot be determined B.{(–3, 4), (–1, 5), (2, 3), (1, –2)} C.{(–3, 4), (–1, 5), (2, 3), (1, 1), (–2, 1)} D.{(4, –3), (5, –1), (3, 2), (1, 1), (1, –2)} GEOMETRY The ordered pairs of the relation {(–3, 4), (–1, 5), (2, 3), (1, 1), (–2, 1)} are the coordinates of the vertices of a pentagon. What is the inverse of this relation?

16. Concept

17. Example 2 Find and Graph an Inverse Step 1Replace f(x) with y in the original equation. Then graph the function and its inverse. Step 2Interchange x and y.

18. Example 2 Find and Graph an Inverse Step 3Solve for y. Inverse Step 4Replace y with f –1 (x). y = –2x + 2  f –1 (x) = –2x + 2 Multiply each side by –2. Add 2 to each side.

19. Example 2 Find and Graph an Inverse

20. Example 2 Find and Graph an Inverse Answer:

21. Example 2 A. B. C. D. Graph the function and its inverse.

22. Concept

23. Example 3 Verify that Two Functions are Inverses Check to see if the compositions of f(x) and g(x) are identity functions.

24. Example 3 Verify that Two Functions are Inverses Answer: The functions are inverses since both [f ○ g](x) and [g ○ f](x) equal x.

25. Example 3 A.They are not inverses since [f ○ g](x) = x + 1. B.They are not inverses since both compositions equal x. C.They are inverses since both compositions equal x. D.They are inverses since both compositions equal x + 1.

26. End of the Lesson