Functions

1. Functions Lesson 2

2. Warm Up • 1. Write an equation of the line that passes through the points (-2, 1) and (3, 2). • 2. Find the gradient of the line that is perpendicular to the line 4x – 7y = 12. • 3. Write the equation of the vertical line that passes through the point (3, 2).

3. Relation • Relation – pairs of quantities that are related to each other • Example: The area A of a circle is related to its radius r by the formula

4. Function • There are different kinds of relations. • When a relation matches each item from one set with exactly one item from a different set the relation is called a function.

5. Definition of a Function • A function is a relationship between two variables such that each value of the first variable is paired with exactly one value of the second variable. • The domain is the set of permitted x values. • The range is the set of found values of y. These will be called images.

6. Let’s take a look at the function that relates the time of day to the temperature.

7. Rules to be a Function

8. Is it a Function? • For each x, there is only one value of y. • Therefore, it IS a function.

9. Is it a function? • Three different y-values (7, 8, and 10) are paired with one x-value. • Therefore, it is NOT a function

10. Function? • Is it a function? Name the domain and range. • YES. For every x-value, there is only one value of y. • Domain: (3, 4, 5, 7, 8) • Range: (-5, -8, 6, 10, 2) {(3, -5), (4, -8), (5, 6), (7, 10), (8, 2)}

11. Function? • Is it a function? State the domain and range. • No. The x-value of 5 is paired with two different y-values. • Domain: (5, 6, 3, 4, 12) • Range: (8, 7, -1, 2, 9, -2) {(5, 8), (6, 7), (3, -1), (4, 2), (5, 9), (12, -2)

12. Function? • Is it a function? Name the domain and range. • Yes. For every x-value, there is only one value of y. • Domain: (-2, 4, 3, 7, 9, 2) • Range: (3, 6, 1, -3, 8) {(-2, 3), (4, 6), (3, 1), (7, 6), (9, -3), (2, 8)}

13. Function? YES

14. Vertical Line Test • Used to determine if a graph is a function. • If a vertical line intersects the graph at more than one point, then the graph is NOT a function.

15. NOT a function

16. IS a function

17. You Try…...

18. You Try….

19. You Try: Is it a Function? • YES

20. You Try…Is it a function? • YES.

21. You Try…Is it a Function? • NO.

22. Is it a function? Give the domain and range.

23. Give the Domain and Range.

24. IB Notation…. • When a function is defined for all real values, we write the domain of f as

25. Functional Notation • We have seen an equation written in the form y = some expression in x. • Another way of writing this is to use functional notation. • For Example, you could write y = x² as f(x) = x².

27. Functional Notation f(x) = 3x + 5 Find:

28. Functional Notation Find:

29. Functional Notation Find:

30. Let’s look at Functions Graphically

31. Find:

32. Find:

33. Find:

34. Find:

35. Find:

36. Find:

37. Find:

38. Find:

39. Piecewise-Defined Function

40. A piecewise-defined function is a function that is defined by two or more equations over a specified domain. • The absolute value function can be written as a piecewise-defined function. • The basic characteristics of the absolute value function are summarized on the next page.

42. Example • Evaluate the function when x = -1 and 0.

43. Domain of a Function

44. The domain of a function can be implied by the expression used to define the function • The implied domain is the set of all real numbers for which the expression is defined. • For example,

45. The function has an implied domain that consists of all real x other than x = ±2 • The domain excludes x-values that result in division by zero.

46. Another common type of implied domain is that used to avoid even roots of negative numbers. • EX: is defined only for The domain excludes x-values that result in even roots of negative numbers.