1. Parametric Relations and Inverses
Demana, Waits, Foley, Kennedy
1.5
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.
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4. Example: Defining a Function Parametrically
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5. Solution t x = t – 1 y = t2 + 2 (x, y) –3 –4 11 (–4, 11) –2 6 (–3, 6)–1 3
(–2, 3) 2
(–1, 2) 1
(0, 3)
(1, 6)
(2, 11)
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6. Example: Defining a Function Parametrically
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7. Solution
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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. y = x is 1:1, but y = sin(x) is not 1:1
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. From the text book: Section 1.5
Suggested HW Problems
From the text book: Section 1.5
Page 126: know 9 – 32, 39 – 44
From Handout 2.8: know