1. Splash Screen

2. 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
Lesson Menu

3. Using f(x) = 3x + 2 and g(x) = 2x2 – 1, find (f – g)(x), if it exists.
A. (f – g)(x) = 2x2 + 3x + 1
B. (f – g)(x) = 2x2 + 3x + 3
C. (f – g)(x) = –x2 – x – 1
D. (f – g)(x) = –2x2 + 3x + 3
5-Minute Check 1

4. Using f(x) = 3x + 2 and g(x) = 2x2 – 1, find (f – g)(x), if it exists.
A. (f – g)(x) = 2x2 + 3x + 1
B. (f – g)(x) = 2x2 + 3x + 3
C. (f – g)(x) = –x2 – x – 1
D. (f – g)(x) = –2x2 + 3x + 3
5-Minute Check 1

5. Using f(x) = 3x + 2 and g(x) = 2x2 – 1, find (f ● g)(x), if it exists.
A. (f ● g)(x) = 6x3 + 4x2 – 3x – 2
B. (f ● g)(x) = 6x2 + 4x – 2
C. (f ● g)(x) = 5x2 + 4x – 3
D. (f ● g)(x) = x2 + 6x + 1
5-Minute Check 2

6. Using f(x) = 3x + 2 and g(x) = 2x2 – 1, find (f ● g)(x), if it exists.
A. (f ● g)(x) = 6x3 + 4x2 – 3x – 2
B. (f ● g)(x) = 6x2 + 4x – 2
C. (f ● g)(x) = 5x2 + 4x – 3
D. (f ● g)(x) = x2 + 6x + 1
5-Minute Check 2

7. Using f(x) = 3x + 2 and g(x) = 2x2 – 1, find [f ○ g](x), if it exists.
A. [f ○ g](x) = 12x2 + 6x + 1
B. [f ○ g](x) = 6x2 + 4x – 2
C. [f ○ g](x) = 6x2 – 1
D. [f ○ g](x) = 3x2 – 2
5-Minute Check 3

8. Using f(x) = 3x + 2 and g(x) = 2x2 – 1, find [f ○ g](x), if it exists.
A. [f ○ g](x) = 12x2 + 6x + 1
B. [f ○ g](x) = 6x2 + 4x – 2
C. [f ○ g](x) = 6x2 – 1
D. [f ○ g](x) = 3x2 – 2
5-Minute Check 3

9. Using f(x) = 3x + 2 and g(x) = 2x2 – 1, find [g ○ f](x), if it exists.
A. [g ○ f](x) = 18x3 + 24x2 + 7x + 1
B. [g ○ f](x) = 18x2 + 24x + 7
C. [g ○ f](x) = 6x2 + 24x + 7
D. [g ○ f](x) = 6x3 + 4x2 – 3x – 2
5-Minute Check 4

10. Using f(x) = 3x + 2 and g(x) = 2x2 – 1, find [g ○ f](x), if it exists.
A. [g ○ f](x) = 18x3 + 24x2 + 7x + 1
B. [g ○ f](x) = 18x2 + 24x + 7
C. [g ○ f](x) = 6x2 + 24x + 7
D. [g ○ f](x) = 6x3 + 4x2 – 3x – 2
5-Minute Check 4

11. 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.
[g ○ f](x) = 0.75x + 15
[g ○ f](x) = 0.75(x + 20) + 15
C. [g ○ f](x) = x + 20(0.75x + 15)
D. [g ○ f](x) = 1.75x + 35
5-Minute Check 5

12. 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.
[g ○ f](x) = 0.75x + 15
[g ○ f](x) = 0.75(x + 20) + 15
C. [g ○ f](x) = x + 20(0.75x + 15)
D. [g ○ f](x) = 1.75x + 35
5-Minute Check 5

13. Let f(x) = x – 3 and g(x) = x2. Which of the following is equivalent to (f ○ g)(1)?
A. f(1)
B. g(1)
C. (g ○ f)(1)
D. f(0)
5-Minute Check 6

14. Let f(x) = x – 3 and g(x) = x2. Which of the following is equivalent to (f ○ g)(1)?
A. f(1)
B. g(1)
C. (g ○ f)(1)
D. f(0)
5-Minute Check 6

15. Mathematical Practices 7 Look for and make use of structure.
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.
CCSS

16. 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.
Then/Now

17. inverse relation
inverse function
Vocabulary

18. Concept

19. 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)}.
Example 1

20. Find an Inverse Relation
Answer:
Example 1

21. 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.
Example 1

22. 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?
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)}
Example 1

23. 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?
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)}
Example 1

24. Concept

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

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

27. Find and Graph an Inverse
Example 2

28. Find and Graph an Inverse
Answer:
Example 2

29. Find and Graph an Inverse
Answer:
Example 2

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

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

32. Concept

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

34. Verify that Two Functions are Inverses
Answer:
Example 3

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

36. 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.
Example 3

37. 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.
Example 3

38. End of the Lesson