Relations

B-Man Relations & Functions: Ordered Pairs, Tables and Graphs

Open Ended

2. Give a brief statement similar to those on the previous slide to

3. describe two quantities that are related.

Relations ~

Many quantities are related. For example:
1. A person's height is related to their age.
2. The number of gallons of water needed on a ranch on a daily basis is related to the number of animals present.
3. The annual income of a person in the USA is related to their education level. (Look it up!)
4. The number of calories in a serving of a particular food is related to the size of the serving.

​Relations may sometimes be shown as ordered pairs:

(#children, #gal. milk)

Ordered Pairs:

​Relations may be shown as a table.

Table:

​A mapping diagram is often useful in depicting a relation.

​​Mapping Diagram

​Relations are often shown as graphs to help visualize the relation.

​​Graph

Domain - set of input (often x) values; think x-coordinate

Range - set of output (often y)
values; think y-coordinate
In the relation shown as ordered pairs at rt:
D: {-2, 1, 4}
R: {-3, 0, 2, 5}
NOTE: Domain and range are listed in INCREASING order from LEFT to RIGHT, and NEVER repeated.

Domain & Range

Multiple Choice

Correctly list the domain for the relation shown here as a set of ordered pairs:

{(-5, 4), (-3, 2), (-3, 4), (3, 7)}

• {-5, -3, -3, 3}

• {2, 4, 4, 7}

• {2, 4, 7}

• {-5, -3, 3}

Multiple Choice

Correctly state the range for the relation shown here as a set of ordered pairs:

{(-5, 4), (-3, 2), (-3, 4), (3, 7)}

• {-5, -3, -3, 3}

• {2, 4, 4, 7}

• {2, 4, 7}

• {-5, -3, 3}

Multiple Choice

Correctly list the range for the relation shown here as a mapping diagram:

• {0, 2, 2, 4}

• {(0,0),(2, 1),(2,-2),(4,3)}

• {0, 2, 4}

• {-3, -2, 0, 1}

Multiple Choice

Correctly list the domain for the relation shown here as a mapping diagram:

• {0, 2, 2, 4}

• {(0,0),(2, 1),(2,-2),(4,3)}

• {0, 2, 4}

• {-3, -2, 0, 1}

Fun with Functions~

Many quantities are related. For example:
1. Heart rate is related to how much a person exerts themself.
2. Weekly wages earned by an hourly worker is related to how many hours the person works in a week.
3. The number gallons of gasoline used on a road trip in a car depends on how many miles the car is driven.
4. The volume of a cylindrical container is related to the diameter AND height of the container.

Functions v. Relations ~

A function is a relation where each input in the domain generates exactly one ouput in the range.
In each of the following cases, discuss with a parnter whether you think the relation described is a function or not? Explain.

1. Heart rate is related to how many minutes a person exercises.

2. Weekly wages earned by an hourly worker is related to how many hours the person works in a week.

3. The number gallons of gasoline used on a road trip in a car depends on how many miles the car is driven.

4. The volume of a cylindrical container is related to the diameter AND height of the container.

THINK: Is it possible for the SAME input (x) value to give DIFFERENT output (y) values?

​Functions may be shown as ordered pairs which can help organize data and look for patterns.

Ordered Pairs:

​Functions may be shown as a table to aid in graphing or organize data to look for patterns.

Table:

​A mapping diagram is often useful in showing a relation is a function or not. Each input has only 1 Arrow!

​​Mapping Diagram

​The graph of a relation can also show if it is a function or not! (Vertical line test.)

​​Graph

​Recap: Function v. Relation

x
-5​
-3​
-1​
​

​y
0​
1​
2​
4​

​x
-4​
-2​
2​
4​

​y
​
1​
​
2​

​From a set of Ordered Pairs:
Focus on the x-coordinates! If ANY x-coordinate is repeated, then the relation is NOT a function.
A = {(-5,0), (-3,1), (-1,2), (-1,4)}
NOT a function since x=-1 has TWO outputs y=2 and y=4.

​NOTE: In the related mapping diagram, the input (domain) element x = -1 has TWO arrows coming from it. Thus, NOT a fucntion.

​From a set of Ordered Pairs:
Focus on the x-coordinates! If EVERY
x-coordinate only occurs ONCE, then the relation IS a function.
B = {(-4,2), (-2,1), (2,1), (4,2)}
IS a function since EVERY x-coordinate has ONLY ONE output for y.

​NOTE: In the related mapping diagram, each input (domain) element x has ONLY ONE arrow coming from it. Thus, IS a fucntion.

Categorize

Options (10)

{(-3,2),(-1,5),(1,7),(3,9)}

{(-9,-3), (-4,-2), (-4,2), (-9,3)}

$$K\left(v\right)=\frac{1}{2}\cdot500v^2$$  

$$x^2+y^2=25$$  

Organize these relations as FUNCTIONS or NOT FUNCTIONS!

FUNCTION

NOT A FUNCTION

Watch the video and think about any relations or functions are discussed in the video.

​Take a break and watch this fascinating video about 200 countries over 200 years in 200 seconds!

Open Ended

Consider the video you just watched. Answer the following:

1. What relation(s) is(are) described in the video?

2. Is the scale on each axis uniformly incremented?

3. Describe something you found interesting in the video?

4. What happened to world population around 1914-1918? What were the main causes?

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