Objectives Identify functions. Find the domain and range of relations and functions.

Relationships can be represented by a set of ordered pairs called a relation. 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.

Example 1: Showing Multiple Representations of 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

Example 1 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.

Example 1 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

Check It Out! Example 1 Express the relation {(1, 3), (2, 4), (3, 5)} as a table, as a graph, and as a mapping diagram. Table x y Write all x-values under “x” and all y-values under “y”.

Check It Out! Example 1 Continued Express the relation {(1, 3), (2, 4), (3, 5)} as a table, as a graph, and as a mapping diagram. Graph Use the x- and y-values to plot the ordered pairs.

Check It Out! Example 1 Continued Express the relation {(1, 3), (2, 4), (3, 5)} 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.

The domain of a relation is the set of first coordinates (or x-values) of the ordered pairs. The range of a relation is the set of second coordinates (or y-values) of the ordered pairs. The domain of the track meet scoring system is {1, 2, 3, 4}. The range is {5, 3, 2, 1}. The domain and range are in alphabetical order. (X,Y) is the same as (D,R)

Example 2: Finding the Domain and Range of a Relation 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

Check It Out! Example 2a Give the domain and range of the relation. 1 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: Range:

x y Check It Out! Example 2b Give the domain and range of the relation. x y 1 4 8 The domain values are all x-values 1, 4, and 8. The range values are y-values 1 and 4. Domain: Range:

Each X can be paired with only one Y!! A function is a special type of relation that pairs each domain value with exactly one range value. Which means… Each X can be paired with only one Y!!

Example 3A: Identifying Functions Give the domain and range of the relation. Tell whether the relation is a function. Explain. {(3, –2), (5, –1), (4, 0), (3, 1)} D: R: 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.

Example 3B: Identifying Functions Give the domain and range of the relation. Tell whether the relation is a function. Explain. –4 2 –8 1 4 5 D: R: This relation is a function. Each domain value is paired with exactly one range value.

Example 3C: Identifying Functions 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 D: R: The relation is not a function. Nearly all domain values have more than one range value.

Check It Out! Example 3 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: R: D: R:

Graphs can be used to illustrate many different situations Graphs can be used to illustrate many different situations. For example, trends shown on a cardiograph can help a doctor see how a patient’s heart is functioning. To relate a graph to a given situation, use key words in the description.

Example 1: Relating Graphs to Situations Each day several leaves fall from a tree. One day a gust of wind blows off many leaves. Eventually, there are no more leaves on the tree. Choose the graph that best represents the situation. Step 1 Read the graphs from left to right to show time passing.

Check It Out! Example 1 The air temperature increased steadily for several hours and then remained constant. At the end of the day, the temperature increased slightly before dropping sharply. Choose the graph that best represents this situation. Step 1 Read the graphs from left to right to show time passing .

As seen in Example 1, some graphs are connected lines or curves called continuous graphs. Some graphs are only distinct points. They are called discrete graphs The graph on theme park attendance is an example of a discrete graph. It consists of distinct points because each year is distinct and people are counted in whole numbers only. The values between whole numbers are not included, since they have no meaning for the situation.

Example 2A: Sketching Graphs for Situations Sketch a graph for the situation. Tell whether the graph is continuous or discrete. A truck driver enters a street, drives at a constant speed, stops at a light, and then continues. As time passes during the trip (moving left to right along the x-axis) the truck's speed (y-axis) does the following: Speed Time y x • initially increases • remains constant • decreases to a stop • increases The graph is continuous.

Example 2B: Sketching Graphs for Situations Sketch a graph for the situation. Tell whether the graph is continuous or discrete. A small bookstore sold between 5 and 8 books each day for 7 days. The number of books sold (y-axis) varies for each day (x-axis). Since the bookstore accounts for the number of books sold at the end of each day, the graph is 7 distinct points. The graph is discrete.

Check It Out! Example 2a Sketch a graph for the situation. Tell whether the graph is continuous or discrete. Jamie is taking an 8-week keyboarding class. At the end of each week, she takes a test to find the number of words she can type per minute. She improves each week. Each week (x-axis) her typing speed is measured. She gets a separate score (y-axis) for each test. Since each score is separate, the graph consists of distinct units. The graph is discrete.

Example 3: Writing Situations for Graphs Write a possible situation for the given graph. Step 1 Identify labels. x-axis: time y-axis: speed Step 2 Analyze sections. over time, the speed: • initially decreases, • remains constant, • and then decreases to zero. Possible Situation: A car approaching traffic slows down, drives at a constant speed, and then slows down until coming to a complete stop.

The graph is continuous. Check It Out! Example 2b Sketch a graph for the situation. Tell whether the graph is continuous or discrete. Henry begins to drain a water tank by opening a valve. Then he opens another valve. Then he closes the first valve. He leaves the second valve open until the tank is empty. As time passes while draining the tank (moving left to right along the x-axis) the water level (y-axis) does the following: Water tank Water Level Time • initially declines • decline more rapidly • and then the decline slows down. The graph is continuous.

Check It Out! Example 3 Write a possible situation for the given graph Step 1 Identify labels. x-axis: students y-axis: pizzas Step 2 Analyze sections. As students increase, the pizzas do the following: • initially remains constant, • and then increases to a new constant. Possible Situation: When the number of students reaches a certain point, the number of pizzas bought increases.

Lesson Quiz: Part I 1. Express the relation {(–2, 5), (–1, 4), (1, 3), (2, 4)} as a table, as a graph, and as a mapping diagram. 2. Give the domain and range of the relation.

Lesson Quiz Contd. 3. Give the domain and range of the relation. Tell whether the relation is a function. Explain.

CW/HW for today: Pages 239-242 #2-20(E), #32-34 (E) Page 243 # 1 – 3 (all) Page 233-235 #4 – 16(E), # 23 – 24(both)