1. Formalizing Relations and Functions
Section 4-6 Part 1

2. Goals Goal Rubric To determine whether a relation is a function.
Level 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced problems.
Level 5 – Adapts and applies the goals to different and more complex problems.

3. Vocabulary
Relation
Domain
Range
Vertical line test
Function notation

4. Definition
Relation – A relationships that can be represented by a set of ordered pairs.
Example:
In the scoring systems of some track meets, for first place you get 5 points, for second place you get 3 points, for third place you get 2 points, and for fourth place you get 1 point. This scoring system is a relation, so it can be shown by ordered pairs. {(1, 5), (2, 3), (3, 2) (4, 1)}.
You can also show relations in other ways, such as tables, graphs, or mapping diagrams.

5. Example: Relations
Express the relation {(2, 3), (4, 7), (6, 8)} as a table, as a graph, and as a mapping diagram.
x y
Table
Write all x-values under “x” and all y-values under “y”. 2 4 6 3 7 8

6. Example: Continued
Express the relation {(2, 3), (4, 7), (6, 8)} as a table, as a graph, and as a mapping diagram.
Graph
Use the x- and y-values to plot the ordered pairs.

7. Example: Continued
Express the relation {(2, 3), (4, 7), (6, 8)} as a table, as a graph, and as a mapping diagram.
Mapping Diagram x y
Write all x-values under “x” and all y-values under “y”. Draw an arrow from each x-value to its corresponding y-value. 2 6 4 3 8 7

8. Your Turn:
Express the relation {(1, 3), (2, 4), (3, 5)} as a table, as a graph, and as a mapping diagram.
Mapping Diagram
x y
Table
Graph x y 1 3 1 3 2 4 2 4 3 5 3 5

9. Definition
Domain – The set of first coordinates (or x-values) of the ordered pairs of a relation.
Example:
For the track meet scoring system relation, {(1, 5), (2, 3), (3, 2) (4, 1)}. The domain of the track meet scoring system is {1, 2, 3, 4}.
Range - The set of second coordinates (or y-values) of the ordered pairs of a relation.
For the track meet scoring system relation, {(1, 5), (2, 3), (3, 2) (4, 1)}. The range is {5, 3, 2, 1}.

10. Example: Finding Domain and Range from a Graph
All y-values appear somewhere on the graph.
y =5x
All x-values appear somewhere on the graph.
For y = 5x the domain is all real numbers and the range is all real numbers.

11. Example: Finding Domain and Range from a Graph
Only nonnegative y-values appear on the graph.
y = x2
All x-values appear somewhere on the graph.
For y = x2 the domain is all real numbers and the range is y ≥ 0.

12. Your Turn: Give the domain and range of the relation.
The domain value is all x-values from 1 through 5, inclusive.
The range value is all y-values from 3 through 4, inclusive.
Domain: 1 ≤ x ≤ 5
Range: 3 ≤ y ≤ 4

13. Your Turn: Give the domain and range of the relation. Range Domain
Domain: all real numbers
Range: y ≥ 0

14. Your Turn: Give the domain and range of the relation. Range
Domain: all real numbers
Range: y > -4
Domain

15. Your Turn: Give the domain and range of the relation. 61 2 6 5
The domain values are all x-values 1, 2, 5 and 6. –4 –1
The range values are y-values 0, –1 and –4.
Domain: {6, 5, 2, 1}
Range: {–4, –1, 0}

16. Your Turn: x y Give the domain and range of the relation. 1 4 8
The domain values are all x-values 1, 4, and 8.
The range values are y-values 1 and 4.
Domain: {1, 4, 8}
Range: {1, 4}

17. Functions
A function is a special type of relation that pairs each domain value with exactly one range value.
All functions are relations, but all relations are not functions.

18. Example:
Give the domain and range of the relation. Tell whether the relation is a function. Explain.
{(3, –2), (5, –1), (4, 0), (3, 1)}
Even though 3 is in the domain twice, it is written only once when you are giving the domain.
D: {3, 5, 4}
R: {–2, –1, 0, 1}
The relation is not a function. Each domain value does not have exactly one range value. The domain value 3 is paired with the range values –2 and 1.

19. Example:
Give the domain and range of the relation. Tell whether the relation is a function. Explain. –4
Use the arrows to determine which domain values correspond to each range value. 2 –8 1 4 5
D: {–4, –8, 4, 5}
R: {2, 1}
This relation is a function. Each domain value is paired with exactly one range value.

20. Example:
Give the domain and range of the relation. Tell whether the relation is a function. Explain.
Draw in lines to see the domain and range values
Range
Domain
D: –5 ≤ x ≤ 3 R: –2 ≤ y ≤ 1
The relation is not a function. Nearly all domain values have more than one range value.

21. Your Turn:
Give the domain and range of each relation. Tell whether the relation is a function and explain.
a. {(8, 2), (–4, 1),
(–6, 2),(1, 9)} b.
D: {–6, –4, 1, 8}
R: {1, 2, 9}
D: {2, 3, 4}
R: {–5, –4, –3}
The relation is a function. Each domain value is paired with exactly one range value.
The relation is not a function. The domain value 2 is paired with both –5 and –4.

22. Your Turn:
Give the domain and range of the relation. Tell whether the relation is a function. Explain.
D: {5, 10, 15};
R: {2, 4, 6, 8};
The relation is not a function since 5 is paired with 2 and 4.

23. Your Turn:
Give the domain and range of each relation. Tell whether the relation is a function and explain.
D: {-2, -1, 1, 2}
R: {3, 4, 5}
The relation is a function. Each domain value is paired with exactly one range value.

24. Vertical line Test
When an equation has two variables, its solutions will be all ordered pairs (x, y) that makes the equation true. Since the solutions are ordered pairs, it is possible to represent them on a graph. When you represent all solutions of an equation on a graph, you are graphing the equation.
Since the solutions of an equation that has two variables are a set of ordered pairs, they are a relation.
One way to tell if this relation is a function is to graph the equation and use the vertical-line test.

25. Vertical Line Test

26. Example:
Graph each equation. Then tell whether the equation represents a function.
–3x + 2 = y
Step 1 Choose several values of x and generate ordered pairs.
Step 2 Plot enough points to see a pattern. 
–3(–1) + 2 = 5
–3(0) + 2 = 2
–3(1) + 2 =–1 –1 1
–3x + 2 = y x
(x, y)
(–1, 5)
(0, 2)
(1, –1)

27. Example: Continued
Step 3 The points appear to form a line. Draw a line through all the points to show all the ordered pairs that satisfy the function. Draw arrowheads on both “ends” of the line.
Step 4 Use the vertical line test on the graph. 
No vertical line will intersect the graph more than once. The equation –3x + 2 = y represents a function. 

28. Example:
Graph each equation. Then tell whether the equation represents a function.
y = |x| + 2
Step 1 Choose several values of x and generate ordered pairs.
Step 2 Plot enough points to see a pattern.
1 + 2 = 3
0 + 2 = 2 –1 1
|x| + 2 = y x
(x, y)
(–1, 3)
(0, 2)
(1, 3) 

29. Example: Continued
Step 3 The points appear to form a V-shaped graph. Draw two rays from (0, 2) to show all the ordered pairs that satisfy the function. Draw arrowheads on the end of each ray.
Step 4 Use the vertical line test on the graph. 
No vertical line will intersect the graph more than once. The equation y = |x| + 2 represents a function.

30. Your Turn:
Graph each equation. Then tell whether the equation represents a function.
y = 3x – 2
Step 2 Plot enough points to see a pattern.
Step 1 Choose several values of x and generate ordered pairs. 
3(–1) – 2 = –5 –1 1
3x – 2 = y x
(x, y)
(–1, –5)
(0, –2)
(1, 1)
3(0) – 2 = –2
3(1) – 2 = 1

31. Your Turn: Continued
Step 3 The points appear to form a line. Draw a line through all the points to show all the ordered pairs that satisfy the function. Draw arrowheads on both “ends” of the line.
Step 4 Use the vertical line test on the graph. 
No vertical line will intersect the graph more than once. The equation y = 3x – 2 represents a function.

32. Your Turn:
Graph each equation. Then tell whether the equation represents a function.
y = |x – 1|
Step 2 Plot enough points to see a pattern.
Step 1 Choose several values of x and generate ordered pairs.
(2, 1)
(–1, 2)
(0, 1)
(1, 0)
(x, y)
y = |x – 1| x
2 = |–1 – 1|
1 = |0 – 1|
0 = |1 – 1|
1 = |2 – 1| –1 1 2 

33. Your Turn: Continued
Step 3 The points appear to form a V-shaped graph. Draw two rays from (1, 0) to show all the ordered pairs that satisfy the function. Draw arrowheads on the end of each ray.
Step 4 Use the vertical line test on the graph.
No vertical line will intersect the graph more than once. The equation y = |x – 1| represents a function. 

34. Example: Determine whether the discrete relation is a function.y 1 2 3 4 1 2 3 4
y is a function of x.

35. Your Turn: y is not a function of x.
Determine whether the discrete relation is a function.
y is not a function of x.

36. Identifying Functions

37. Determine if the relationship represents a function.1. x y 2 3 4 5 6
The input x = 2 has two outputs, y = 3 and y = 6. The input x = 3 also has more than one output.
The relationship is not a function.

38. Determine if the relationship represents a function.2. x y –1 2 –4 5
– 7 8
–10
Each input has only one output value.
The relationship is a function.

39. Determine if the relationship represents a function.3.
Pass a vertical line across the graph. Many vertical lines intersect the graph at two points.
The relationship is not a function.

40. Determine if the relationship represents a function. 4.x y 2 -2 -4 4
Pass a vertical line across the graph. No vertical lines intersect the graph at more than one point.
The relationship is a function.

41. Determine if the relationship represents a function.5. x y 1 2 3
Each input x has only one output y.
The relationship is a function.

42. Determine if the relationship represents a function.6. x y 2 1 3 4 5
Each input has only one output value.
The relationship is a function.

43. Determine if the relationship represents a function. 7.x y
Pass a vertical line across the graph. No vertical lines intersect the graph at more than one point. 2 -2 2
The relationship is a function. -2

44. Determine if the relationship represents a function. 8.x y 2 -2 4 -4
Pass a vertical line across the graph. No vertical lines intersect the graph at more than one point.
The relationship is a function.

45. Function Notation
An algebraic expression that defines a function is a function rule.
If x is the independent variable and y is the dependent variable, then function notation for y is f(x), read “f of x,” where f names the function.
When an equation in two variables describes a function, you can use function notation to write it.

46. y = f(x) Function Notation
The dependent variable is a function of the independent variable.
y is a function of x.
y = f (x)
y = f(x)

47. Inputs & Outputs Function Input Output
The x values, can be thought of as the inputs and the y values or f (x), can be thought of as the outputs.
Function
Input
Output
f (x) f x
Independent
variable
Dependent
variable

48. Example: Evaluating Functions
Evaluate the function for the given input values.
For f(x) = 3x + 2, find f(x) when x = 7 and when x = –4.
f(x) = 3(x) + 2
f(x) = 3(x) + 2
Substitute
7 for x.
Substitute
–4 for x.
f(7) = 3(7) + 2
f(–4) = 3(–4) + 2 =
= –12 + 2
Simplify.
Simplify.
= 23
= –10

49. Example: Evaluating Functions
Evaluate the function for the given input values.
For g(t) = 1.5t – 5, find g(t) when t = 6 and when t = –2.
g(t) = 1.5t – 5
g(t) = 1.5t – 5
g(6) = 1.5(6) – 5
g(–2) = 1.5(–2) – 5
= 9 – 5
= –3 – 5
= 4
= –8

50. Example: Evaluating Functions
Evaluate the function for the given input values.
For , find h(r) when r = 600 and when r = –12.
= 202
= –2

51. Your Turn: Evaluate the function for the given input values.
For h(c) = 2c – 1, find h(c) when c = 1 and when c = –3.
h(c) = 2c – 1
h(c) = 2c – 1
h(1) = 2(1) – 1
h(–3) = 2(–3) – 1
= 2 – 1
= –6 – 1
= 1
= –7

52. Your Turn: Evaluate each function for the given input values.
For g(t) = , find g(t) when t = –24 and when t = 400.
= –5
= 101

53. Example: Evaluating Functions from a Graph
Use a graph of the function to find the value of f(x) when x = –4.
Locate –4 on the x-axis. Move up to the graph of the function. Then move right to the y-axis to find the corresponding value of y.
f(–4) = 6

54. Your Turn: Use the graph of to find the value of x when f(x) = 3.
Locate 3 on the y-axis. Move right to the graph of the function. Then move down to the x-axis to find the corresponding value of x.
f(3) = 3

55. Joke Time How can you get four suits for a dollar?
Buy a deck of cards.
Guy buys a parrot that is constantly using foul language. Really horrible stuff. Finally the guy gets fed up and throws the parrot in the freezer to punish him. After about an hour, he hears a faint tapping sound from inside the freezer and opens the door. There’s the parrot, wings wrapped around himself, shivering. He says,
“I swear, I’ll never, ever curse again. But can I ask you a question? What did the chicken do?"

56. Assignment
4-6 Part 1 Exercises Pg. 290: #8 – 20 even