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Parametric Relations and Inverses

Learn about relations defined parametrically and inverse relations and functions. Understand why some functions are best defined parametrically and others as inverses of known functions.

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Parametric Relations and Inverses

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  1. 1.5 Demana, Waits, Foley, Kennedy Parametric Relations and Inverses

  2. What you’ll learn about • Relations 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. 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. Skip this for now

  4. Example: Defining a Function Parametrically Skip this for now

  5. Solution Skip this for now

  6. Example: Defining a Function Parametrically Skip this for now

  7. Solution Skip this for now

  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. Properties of Inverses List all properties of inverses that you remember from Alg 2

  10. 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.

  11. All of these are functions, find at least two different ways to classify them. • A B C D • E F G H

  12. Inverse Functions A function is considered 1:1 iff it passes both the horizontal and vertical line tests. y = x is 1:1, but y = sin(x) is not 1:1

  13. Example: Finding an Inverse Function Algebraically

  14. Solution

  15. 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.

  16. Example: Finding an Inverse Function Graphically The graph of a function y = f (x) is shown. Sketch a graph of the function y = f –1(x).Is f a one-to-one function?

  17. Solution All we need to do is to find the reflection of the given graph across the line y = x.

  18. Solution All we need to do is to find the reflection of the given graph across the line y = x.

  19. The Inverse Composition Rule

  20. Example: Verifying Inverse Functions

  21. Solution

  22. How to Find an Inverse Function Algebraically

  23. Suggested HW Problems From the text book: Section 1.5 Page 126: know 9 – 32, 39 – 44 From Handout 2.8: know 1 - 50

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