1. Relations Relation: a set of ordered pairs Domain: the set of x-coordinates, independent Range: the set of y-coordinates, dependent When writing the domain and range, do not repeat values.

2. Relations Given the relation: {(2, -6), (1, 4), (2, 4), (0,0), (1, -6), (3, 0)} State the domain: D: {0,1, 2, 3} State the range: R: {-6, 0, 4}

3. Relations Relations can be written in several ways: ordered pairs, table, graph, or mapping. We have already seen relations represented as ordered pairs.

4. Table {(2, -6), (1, 4), (2, 4), (0, 0), (1, -6), (3, 0)} The ordered pairs should line up right next to each other

5. Graphing {(2, -6), (1, 4), (2, 4), (0, 0), (1, -6), (3, 0)} Plot the ordered pairs

6. Mapping Create two ovals with the domain on the left and the range on the right. Elements are not repeated. Connect elements of the domain with the corresponding elements in the range by drawing an arrow.

7. Mapping {(2, -6), (1, 4), (2, 4), (0, 0), (1, -6), (3, 0)} 21032103 -6 4 0

8. Functions Functions are relations that have exactly One output (y), dependent variable, for every input, independent variable (x) the members of the domain (x-values) DO NOT repeat. y-values, the range, can be repeated.

9. Do the ordered pairs represent a function? {(3, 4), (7, 2), (0, -1), (-2, 2), (-5, 0), (3, 3)} No, 3 is repeated in the domain. When you input a 3, you can get a 4 or 3 out. {(4, 1), (5, 2), (8, 2), (9, 8), (-4,3), (0,0)} Yes, no x-coordinate is repeated. For each x there is only 1 y that is output.

10. Graphs of a Function Vertical Line Test: If a vertical line is passed over the graph and it intersects the graph in exactly one point, the graph represents a function.

11. x y x y Does the graph represent a function? Name the domain and range. Yes D: all reals R: all reals Yes D: all reals R: y ≥ -4

12. x y x y Does the graph represent a function? Name the domain and range. No D: x ≥ 1 R: all reals No D: all reals R: all reals

13. Does the graph represent a function? Name the domain and range. Yes D: all reals R: y ≥ -6 No D: x = 2 R: all reals x y x y

14. 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 pronounced ‘f’ of ‘x’. f(x) is the dependent variable, (output) The ‘f’ names the function, the ‘x’ tells the independent variable that is being used.

15. Function Notation f(x) is the output or dependent variable We can Evaluate a function when we have an input We can then find the output

16. Value of a Function Since the equation y = 3x + 4 represents a function, we can also write it as f(x) = 3x + 4 Find f(2): f(2) = 3(2) + 4 f(2) = 6 + 4 f(2) = 10 The valve of output when x is 2

17. Value of a Function If f(x) = 2x, find f(-3). f(-3) = 2(-3) =-6 f(-3) = -6

18. Value of a Function If f(x) = x 2 + 3, find f(-4). f(-4) = (-4) 2 + 3 f(-4) = 16 + 3 f(-4) = 19

19. Operations with functions (f+g)(x) means to add the rule part of functions f(x) plus g(x) (f-g)(x) means to subtract the rule part of functions f(x) minus g(x)

20. Operations with functions (f g)(x) means to multiply the rule part of functions f(x) times g(x) ( )(x) means to divide the rule part of functions f(x) divided by g(x)

21. Operations with functions Let f(x) = and g(x) = 1. (f + g)(x) = (f + g)(x) =

22. Operations with functions Let f(x) = and g(x) = 2. (f / g)(x) = (f / g)(x) =

23. Operations with functions Let f(x) = and g(x) = 3. (f – g)(x) = (f – g)(x) =

24. Operations with functions Let f(x) = and g(x) = 3. (f * g)(x) = (f * g)(x) =