1. Algebra 1 Glencoe McGraw-Hill Malinda Young Relations, Functions & Function Notation

2. A relation is a set of ordered pairs. A relation can be represented by a graph, a table, or a mapping.

3. Ordered Pairs Table Graph Mapping (4, 3) (-2, -1) (-3, 2) (2, -4) (0, -4) xy 4 3 -2 -3 2 2-4 0 4 -2 -3 2 0 3 2 -4

4. xy Express the relation { (-1, 0), (2, -4), (-3, 1), (4, -3)} as a table, a graph, and a mapping. Then determine the domain and range. xy 0 2-4 -3 1 4 2 -3 4 0 -4 1 -3 You should recall that the domain is the set of all inputs (x) and the range is the set of all outputs (y). Domain: {-1, 2, -3, 4} Range: {0, -4, 1, -3}

5. The inverse of any relation can be found by switching the inputs and the outputs of each ordered pair. The domain of the relation becomes the range of the inverse. The range of the relation becomes the domain of the inverse.

6. 5 7 4 0 1 2 -9 1 2 -9 5 7 4 0 RelationInverse of the Relation (5, 1) (7, 2) (4, -9) (0, 2) (1, 5) (2, 7) (-9, 4) (2, 0) (5, 1) (7, 2) (4, -9) (0, 2)

7. How does the graph of a relation compare with the graph of the inverse of the relation? Graph the points of this relation, then connect the points: (4, 3) (-2, -1) (-3, 2) (2, -4) (0, -4) (3, 4) (-1, -2) (2, -3) (-4, 2) (-4, 0) Graph the points of the inverse of the relation, then connect the points: Write an observation in your notes.

8. A relation is any set of ordered pairs (x, y). A function is a rule that establishes a relationship between two quantities, called the input and the output. For each input, there is exactly one output – even though two different inputs may give the same output. Huh?

9. That is what’s meant by “even though different inputs” (medium Coke, medium Sprite) “may give the same output” (same price for both items). You notice also that other medium sodas like Sprite also cost the same as a medium Coke. Let’s say you go to Baja Fresh. With your lunch you order a medium Coke. Will you see two different prices for a medium Coke on the menu? No. That is what’s meant by “for each input” (medium Coke), “there is exactly one output” (one price for the medium Coke).

10. One way to look at functions is to make an input/output chart like the one below. This relation is a function. For every x-value (input) there is exactly one y-value (output). xy 1 0 2-4 3 1 4-3 510 612 Is this relation a function?

11. No, it is not a function. Why? Because the input 2 had two different outputs. Think of it like this: 2 Cokes can’t be $4 and $10. xy 1 2 24 1 2 48 210 612 Is this relation a function?

12. Domain: {1, 2, 3, 4, 5, 6} Range: {4, 6, 8, 10, 12, 14} xy 1 4 26 3 8 410 512 614 Does the table represent a function? Explain. If it is a function, name the domain and the range. The table DOES represent a function because for every input there is only one output.

13. Domain: {-3, -2, -1} Range: {4, 6, 8} The table DOES represent a function because for every input there is only one output. xy -34 -26 8 -34 -26 8 Does the table represent a function? Explain. If it is a function, name the domain and the range.

14. xy 53 75 20 97 54 75 No, it is not a function. Why? Because the input 5 had two different outputs. Think of it like this: 5 Cokes can’t be $3 and $4.

15. Does the table represent a function? Explain. If it is a function, name the domain and the range. xy -38 -26 4 02 14 26 Domain: {-3, -2, -1, 0, 1, 2} Range: {2, 4, 6, 8} The table DOES represent a function because for every input there is only one output.

16. Function Notation: When a function is defined by an equation, it is often convenient to name the function. Just as x is commonly used as a variable, the letter f is commonly used to name a function. To use function notation, replace y with f(x). The symbol f(x) is read as “the value of f at x” or simply “f of x”. It does not mean f times x. x-y notationfunction notation

17. Any letter can be used to name a function. If you see f(x), g(x), h(x) or any other variable it is the name for the function and it is used in place of y. In a function, x represents the elements of the domain and f(x) represents the elements of the range.

18. If f(x) = 2x + 3, find the value of the function. You are being asked to replace the x in the function with -6. Replace x with -6. Multiply. Add. In this function, when x = -6, f(x) = -9.

19. If f(x) = 2x + 3, find the value of the function. You are being asked replace the x in the function with 11. Replace x with 11. Multiply. Add. In this function, when x = 11, f(x) = 25.

20. If f(x) = x 2 – 7x + 3, find the value of the function. You are being asked to replace the x in the function with 1. Replace x with 1. Multiply. Add. In this function, when x = 1, f(x) = -3.

21. If g(x) = x 2 + x + 3, find the value of the function. You are being asked to replace the x in the function with 4n. Replace x with 4n. Multiply. In this function, when x = 4n, g(x) = 16n 2 + 4n + 3

22. If h(x) = x 2 + 1 find the value of the function. You are being asked to replace the x in the function with n. Replace x with n. In this function, when x = n, h(x) = n 2 + 1

23. If f(x) = 5x + 2, find the value of the function. You are being asked to replace the x in the function with r. The 4 outside the brackets means that the entire function will be multiplied by 4. Replace x with r. In this function, when x = r, 4[f(r)] = 20r + 8 Multiply all terms by 4.

24. A vertical line test is used to determine whether a graph represents a function. A graph is a function if any vertical line intersects the graph at no more than one point. x (input) y (output) The graph is a function.

25. You can use your pencil to check if a graph is a function. Keep your pencil straight to represent a vertical line and pass it across the graph. If it touches the graph at more than one point, the graph is not a function. x (input) y (output)

26. x (input) y (output) The graph is NOT a function.

27. x (input) y (output) The graph is NOT a function.

28. x (input) y (output) The graph IS a function.

29. x (input) y (output) The graph is NOT a function.

30. x (input) y (output) The graph is a function.

31. x (input) y (output) The graph is a function. Determine whether the relation is a function. y = 2x + 1 xy -2 0 1 2..... -3 1 3 5