1. Exercise Give the domain and range of the following relation.

2. Exercise Give the domain and range of the following relation.
B = {(1, 2), (3, 4), (1, 6)}
D = {1, 3}; R = {2, 4, 6}

3. Exercise Give the domain and range of the following relation.

4. Exercise Give the domain and range of the following relation.

5. Exercise
Is the following set a relation?
E = {1, 2, 3, 4} no

6. Exercise
Is the following set a relation?
F = {1, 2, 3} no

7. Function
A function is a relation in which no two ordered pairs have the same first coordinate.

8. Circle Mapping
B = {(−2, 1), (0, 7), (3, 5), (4, 3)}

9. Circle Mapping −

10. A relation is a function if the circle mapping has only one arrow coming from each element in the domain.

11. Example 1
Make a circle mapping of relation C, and give the domain and range. Is the relation a function? C = {(−3, −2), (−2, 3), (1, −3), (3, 4), (4, 3)}
D = {−3, −2, 1, 3, 4}; R = {−3, −2, 3, 4}

12. Example 1
−3 −
−3 −2 3 4
Yes, relation C is a function.

13. Example 2
Determine whether the relation is a function. Explain your answer.
M = {(1, 5), (3, 4), (2, 3), (3, 3)}
No. Two ordered pairs have the same x-coordinate.

14. Example 2
Determine whether the relation is a function. Explain your answer.
N = {(−1, 4), (2, 5), (3, −3), (3, 5)}
No. Two ordered pairs have the same x-coordinate.

15. Example 2
N = {(−1, 4), (2, 5), (3, −3), (3, 5)}
−1 2 3
−3 5 4

16. Example 3 Is P = {(−3, 6), (1, 6), (2, 6)} a function?
Yes. No two points have the same first coordinate.

17. The letters f and g often denote functions.

18. f = {(−3, 4), (0, 3), (3, −2)}
Domain of f = {−3, 0, 3}
Range of f = {−2, 3, 4}

19. Each element of the range is called a value of the function.

20. f = {(−3, 4), (0, 3), (3, −2)}
Because the ordered pair (0, 3) is a part of the function, we say that the value of function f at 0 is 3.

21. Standard function notation: f (0) = 3
f of zero equals three.

22. For ordered pairs (x, y) the value of f at x is y, written f(x) = y.

23. Example 4
Find the following values for function g = {(−1, 1), (0, 2), (1, 3)}.
g(1)
g(1) = 3, since 1 is paired with 3.

24. Example 4
Find the following values for function g = {(−1, 1), (0, 2), (1, 3)}.
g(0)
g(0) = 2, since 0 is paired with 2.

25. Example 4
Find the following values for function g = {(−1, 1), (0, 2), (1, 3)}.
g(−1)
g(−1) = 1, since −1 is paired with 1.

26. Example 5 Find the following values for function f(x) = 2x + 4. f(3)

27. Example 5 Find the following values for function f(x) = 2x + 4. f(−3)

28. Example 6
Find the range of function g(x) = 3x − 2 for the domain {−3, 0, 2}. Then write the function as a set of ordered pairs.
Find g(x) for each value of the domain.

29. Example 6 g(−3) = 3(−3) − 2 = −9 − 2 = −11 g(2) = 3(2) − 2 = 6 − 2
= 4
= 0 − 2
= −2

30. Example 6 The range of g(x) is {−11, −2, 4}.
The function is g = {(−3, −11), (0, −2), (2, 4)}.

31. Vertical Line Test for Functions
If 2 or more points of the relation lie on the same vertical line, the relation is NOT a function.

32. y x
Relation A

33. y x
Relation B

34. y
Relation C x

35. y
Relation D x