1. Copyright © 2011 Pearson, Inc. 1.5 Parametric Relations and Inverses

2. Copyright © 2011 Pearson, Inc. Slide 1.5 - 2 What you’ll learn about Relations Defined Parametrically Inverse Relations and Inverse Functions … and why Some functions and graphs can best be defined parametrically, while some others can be best understood as inverses of functions we already know.

3. Copyright © 2011 Pearson, Inc. Slide 1.5 - 3 Relations Defined Parametrically Another natural way to define functions or, more generally, relations, is to define both elements of the ordered pair (x, y) in terms of another variable t, called a parameter.

4. Copyright © 2011 Pearson, Inc. Slide 1.5 - 4 Example Defining a Function Parametrically

5. Copyright © 2011 Pearson, Inc. Slide 1.5 - 5 Solution tx = t – 1y = t 2 + 2(x, y) –3–411(–4, 11) –2–36(–3, 6) –1–23(–2, 3) 0–12(–1, 2) 103(0, 3) 216(1, 6) 3211(2, 11)

6. Copyright © 2011 Pearson, Inc. Slide 1.5 - 6 Example Defining a Function Parametrically

7. Copyright © 2011 Pearson, Inc. Slide 1.5 - 7 Solution

8. Copyright © 2011 Pearson, Inc. Slide 1.5 - 8 Inverse Relation The ordered pair (a,b) is in a relation if and only if the pair (b,a) is in the inverse relation.

9. Copyright © 2011 Pearson, Inc. Slide 1.5 - 9 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.

10. Copyright © 2011 Pearson, Inc. Slide 1.5 - 10 Inverse Function

11. Copyright © 2011 Pearson, Inc. Slide 1.5 - 11 Example Finding an Inverse Function Algebraically

12. Copyright © 2011 Pearson, Inc. Slide 1.5 - 12 Solution

13. Copyright © 2011 Pearson, Inc. Slide 1.5 - 13 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.

14. Copyright © 2011 Pearson, Inc. Slide 1.5 - 14 Example Finding an Inverse Function Graphically Graph y = x 3 – 4 and its inverse. Is y a one-to-one function?

15. Copyright © 2011 Pearson, Inc. Slide 1.5 - 15 Solution Graph y = x 3 – 4 and its inverse. Is y a one-to-one function? To find an equation for the inverse, interchange x and y, then solve for y.

16. Copyright © 2011 Pearson, Inc. Slide 1.5 - 16 Solution Note that y 1 and y 2 are reflections of each other about the line y = x. Since the graph passes the horizontal and vertical line tests, f is a one-to-one function.

17. Copyright © 2011 Pearson, Inc. Slide 1.5 - 17 The Inverse Composition Rule

18. Copyright © 2011 Pearson, Inc. Slide 1.5 - 18 Example Verifying Inverse Functions

19. Copyright © 2011 Pearson, Inc. Slide 1.5 - 19 Solution

20. Copyright © 2011 Pearson, Inc. Slide 1.5 - 20 How to Find an Inverse Function Algebraically

21. Copyright © 2011 Pearson, Inc. Slide 1.5 - 21 Quick Review

22. Copyright © 2011 Pearson, Inc. Slide 1.5 - 22 Quick Review Solutions