Understanding Relations and Functions: Domain, Range, and Notation

1. Domain and Range

2. Relations and Functions • We saw yesterday that every relationship between x- and y-values represent a relation. • That means every graph on a coordinate grid represents a relation

3. Relations and Functions • How can we write thisrelation down? • As ordered pairs,a table, or a mappingdiagram. • {(1,5),(3,5),(4,2),(6,-2)} • Is it a function?

4. Relations and Functions • How can we writethis relation down? • We cannot write every point on this graph down – there is always another in between so the only way is to write an equation. • y = 2x + 4 • Is this a function?

5. Domain and Range from Graph in set notation. • D: {1,3,4,6} • R: {-2,2,5}

6. Domain and Range from Graph in set notation. • D: x can be anything • R: y can be anything • A better way is to use set notation

7. To use the symbols of algebra, we could write the domain as Does that look like a foreign language? Let’s translate:

8. The curly braces just tell us we have a set of numbers.

9. The x reminds us that our set contains x-values.

10. The colon says, such that

11. The symbol that looks like an e (or a c sticking its tongue out) says, belongs to or is an element of. . .

12. And the cursive, or script, R is short for the set of real numbers.

13. So we read it, “The set of x such that x belongs to R, the set of real numbers.”

14. Read this: “The set of y, such that y belongs to R, the set of real numbers.”

15. It is not always true that the domain and range can be any real number. Sometimes mathematicians want to study a function over a limited domain.

16. Domain and Range from Graph in set notation. • What do you think of the domain? • What about the range?

17. Domain and Range from Graph in set notation. • What do you think of the domain? • What about the range? • Function or not?

18. Domain and Range from Graph in set notation. • What do you think of the domain? • What about the range? • Function or not?

19. HW Worksheet Domain and Range

20. Function Notation • When we know that a relation is a function, the “y” in the equation can be replaced with f(x). • f(x) is simply a notation to designate a function. It is pronounced ‘f’ of ‘x’. • The ‘f’ names the function, the ‘x’ tells the variable that is being used.

21. Value of a Function Since the equation y = x - 2 represents a function, we can also write it as f(x) = x - 2. Find f(4): f(4) = 4 - 2 f(4) = 2

22. Value of a Function If g(s) = 2s + 3, find g(-2). g(-2) = 2(-2) + 3 =-4 + 3 = -1 g(-2) = -1

23. Value of a Function If h(x) = x2 - x + 7, find h(2c). h(2c) = (2c)2 – (2c) + 7 = 4c2 - 2c + 7

24. Value of a Function If f(k) = k2 - 3, find f(a - 1) f(a - 1)=(a - 1)2 - 3 (Remember FOIL?!) =(a-1)(a-1) - 3 = a2 - a - a + 1 - 3 = a2 - 2a - 2

25. Homework • pg 635 #2, 4, 6, 8 (no sketch)

26. Does the equation represent a function? • 2x + 4y = 8 • y = -0.5x + 2 • This equation produces one output for every input so it is a function • Solve the equation for y. • Substitute any value for x and find how many answers it produces for y. • One: function • More than one: not a function

27. Another example: This equation will produce two outputs for every input and is therefore not a function

28. Function operations • If and find:

29. Function operations • If and find:

30. Function operations • If and find:

31. Homework • Worksheet

32. Domain and Range from equation

33. Inverse Functions • Finding and Graphing