1. 1.51.5 Inverse Functions and Modeling

2. Quick Review

3. Quick Review Solutions

4. What you’ll learn about Inverse Relations Inverse Functions … and why Some functions and graphs can best be understood as inverses of functions we already know. Using a function to model a variable under observation in terms of another variable often allows one to make predictions in practical situations, such as predicting the future growth of a business based on data.

5. Inverse Relation The ordered pair (a,b) is in a relation if and only if the pair (b,a) is in the inverse relation.

6. Horizontal Line Test The inverse of a relation is a function if and only if each horizontal line intersects the graph of the original relation in at most one point.

7. Inverse Function

12. How to Find an Inverse Function Algebraically

15. Example Finding an Inverse Function Algebraically

17. The Inverse Reflection Principle The points (a,b) and (b,a) in the coordinate plane are symmetric with respect to the line y=x. The points (a,b) and (b,a) are reflections of each other across the line y=x.

18. The Inverse Composition Rule

19. Example Verifying Inverse Functions

21. Example A Maximum Value Problem

30. Example Finding the Model and Solving Grain is leaking through a hole in a storage bin at a constant rate of 5 cubic inches per minute. The grain forms a cone-shaped pile on the ground below. As it grows, the height of the cone always remains equal to its radius. If the cone is one foot tall now, how tall will it be in one hour?

31. Example Finding the Model and Solving Grain is leaking through a hole in a storage bin at a constant rate of 5 cubic inches per minute. The grain forms a cone-shaped pile on the ground below. As it grows, the height of the cone always remains equal to its radius. If the cone is one foot tall now, how tall will it be in one hour?

32. Functions

33. Functions (cont’d)

34. Chapter Test

36. Chapter Test Solutions