1. Relations and Functions
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Chapter 3
1.
Relations and Functions
This lesson delves into the mathematical concepts of relations and functions. It explains how to identify whether a given relation is a function using methods like the vertical line test. The domain, which is the set of all possible inputs, and the range, the set of all possible outputs, are also discussed. Practical examples include analyzing ages and heights in a family to determine if they form a function. The vertical line test is highlighted as a graphical method to test if a relation is a function. The lesson is beneficial for students, educators, and anyone interested in understanding the mathematical relationships between sets of numbers.
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Lesson Settings & Tools
| | 11 Theory slides |
| | 11 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Relations and Functions
Slide of 11
When elements of one set are connected somehow with elements of another set, there is a relation between the sets. This lesson will present the formal definition of a relation and its different representations. Also, a particular type of relation, called a function, will be explored in more depth.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
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Explore
Comparing Graphs
Consider the following pair of graphs. Apart from the shape of the graphs, what difference can be established between them? If needed, press the button to show a vertical line. Then, move the line horizontally and watch for the number of times it intersects each graph along the way.
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Discussion
Relations and Mapping Diagrams
In the previous exploration, each graph represented a certain relationship between the variables x and y. In fact, the value of y depends on the value of x. Next, the definition of relation is developed along with one way of visualizing it.
Concept
Relation
A relation, or relationship, is a rule that relates the elements of one set to the elements of another set. The first set is called the set of inputs and the second set is called the set of outputs. In a relation, an input can have zero outputs, one output, or more than one output.
Often, a relation is thought of as a set of ordered pairs of the form (x,y). In this case, the x-values represent the inputs and the y-values the outputs. However, a relation can also be represented using a mapping diagram, a table of values, or a set of points on a coordinate plane.
Concept
Mapping Diagram
A mapping diagram is a graphic tool that helps to visualize a relation. In a mapping diagram, the inputs are listed in one set and the outputs in another. Then, arrows are drawn from each input to its corresponding outputs.
![A mapping diagram representing the relation {(-2,-8),(-1,11),(0,5),(2,-8)} from the input set [-2,-1,0,2] to the output set [-8,11,5].](/mediawiki/images/0/08/Mljsx_Concept_Representing_Relations_3.svg)
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Example
Representing a Relation in Different Ways
Ignacio, the younger brother of Kriz, loves watching Kriz's college volleyball games. He recorded Kriz and their teammates' names, ages, and heights. Ignacio is such a curious kid that he wants to describe some relations based on these values. Then he will share them with his big sibling.
Represent the relation between Kriz's teammates' ages and heights using a table of values, a mapping diagram, a set of ordered pairs, and points in a coordinate plane.
Answer
Table of Values:
| Ages | 18 | 20 | 23 | 20 | 21 | 19 |
|---|---|---|---|---|---|---|
| Heights (m) | 1.70 | 1.96 | 1.85 | 1.75 | 1.91 | 1.87 |
Mapping Diagram:
Ordered Pairs: {(18,1.70), (20,1.96), (23,1.85), (20,1.75), (21,1.91), (19,1.87)}
Coordinate Plane:
Hint
To make a mapping diagram, place all the ages in one set and all the heights in another set. Then, connect each age with its corresponding height using an arrow. To write the relation as a set of ordered pairs, place the ages in the first component and the corresponding heights in the second component.
Solution
Using the information that Ignacio gathered, a relation between the ages and heights can be made and represented using different visualizations.
Table of Values
First, organize the relation using a table of values. Place the ages in the first row and the heights in the second row.
| Ages | 18 | 20 | 23 | 20 | 21 | 19 |
|---|---|---|---|---|---|---|
| Heights (m) | 1.70 | 1.96 | 1.85 | 1.75 | 1.91 | 1.87 |
Mapping Diagram
The same relation can be displayed using a mapping diagram. Place the ages in a set and the heights in a different set. Then, connect each age with its corresponding height by using arrows. It is not necessary to write 20 twice in the ages set.
Set of Ordered Pairs
To write the relation using ordered pairs, the ages will be placed in the first coordinate of each ordered pair, and the corresponding heights will be placed in the second coordinate. { &(18,1.70),(20,1.96),(23,1.85), &(20,1.75),(21,1.91),(19,1.87) }
Points in the Coordinate Plane
Finally, the relation between the ages and heights of the players can be represented using points in the coordinate plane. To do so, the ages will be placed along the horizontal axis and the heights along the vertical axis. Here, the axes will be drawn using different scales.
Note that by hovering over each point, the corresponding information is shown.
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Discussion
Functions, Domain, and Range
Among all the possible relations, those assigning only one output to every input are of particular interest.
Concept
Function
A function is a relation in which each input is assigned to exactly one output. The set of all possible inputs is called the domain of the function and the set of all possible outputs is called the range. If x represents the inputs and y the outputs of a function, it is often said that y is a function of x
or that y depends on x.
y = f(x)
This way of representing the dependent variable is called function notation. A function can be represented using a table, a mapping diagram, an equation, or a graph.
| Determining Whether a Relation Is a Function | |
|---|---|
| If represented as | Use |
| a set of coordinates or a table of values | a mapping diagram |
| a graph in the coordinate plane | the vertical line test |
Concept
Domain
The domain of a function is the set of all x-values, called inputs, for which the function is defined. As an example, consider the following functions. f(x) & = 3x [0.2cm] g(x) & = sqrt(x) [0.3em] h(x) & = 1/x Their domains can be written by analyzing the definition of each function.
| Function | Analysis | Domain |
|---|---|---|
| f(x) = 3x | Multiplying by 3 is defined for all real numbers. | All real numbers |
| g(x) = sqrt(x) | Square roots are not defined for negative numbers. | All non-negative numbers — that is, x≥ 0 |
| h(x) = 1/x | Dividing by zero is undefined. | All real numbers except 0 — that is, x≠ 0 |
The domain of a function can be determined through a variety of methods depending on how the function is represented.
Concept
Range
The range of a function is the set of all y-values, called outputs, of the function. The range depends on both the domain and the function itself. For example, consider the following functions and their defined domains.
| Function | Domain |
|---|---|
| f(x)=2x | All integers |
| g(x)=x^2 | All real numbers |
| h(x)=4 | All real numbers |
The ranges of each function can be determined by analyzing the definition of each function along with the given domains.
| Function | Domain | Analysis | Range |
|---|---|---|---|
| f(x) = 2x | All integers | The function takes any integer input and produces an output that is an even number, as each input is multiplied by 2. | All even numbers |
| g(x) = x^2 | All real numbers | The function takes any real number input and produces an output that is a non-negative number, as each input is squared. | All non-negative numbers. That is, y≥ 0 |
| h(x) = 4 | All real numbers | The function takes any real number input and sends it to 4. | Only the number 4. That is, the range is {4} |
The method used to determine the range of a function can vary depending on how that function is represented.
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Illustration
Viewing Functions as Machines
A comparison for a function is to think of it as a machine. The inputs are the materials placed in the machine, and the outputs are the objects created. In the following applet, there are four preset inputs. This particular machine only accepts numbers between -100 and 100 to be plugged in as materials. See what happens!
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Discussion
Methods to Determine Whether a Relation Is a Function
It is essential to determine whether a relation describes a function or not. For example, imagine that a programmer wrote the following relation for a printer's software.
If a user selects color=0, then they cannot be sure if the document will be printed in black or red. In this case, the programmer needs to write a relation that is a function. Determining if a relation is a function can be done by using either a mapping diagram or the Vertical Line Test. The appropriate method to be used depends on how the relation is represented.
Method
Determining if a Relation Is a Function Using Mapping Diagrams
Given a relation, a mapping diagram can be used to determine whether the relation is a function. For example, consider the relation given by the following set of coordinates. {(4,3),(0,-5),(-3,3),(8,0),(4,-5)} To figure out if a relation is a function, there are four steps to follow.
1
List the Inputs
expand_more Start by writing all the inputs of the relation in a column. The inputs are the first components of each pair if the relation is given as a set of coordinates.
{( 4,3),( 0,-5),( -3,3),( 8,0),( 4,-5)}
Notice that the number 4 is an input for two different points. In the mapping diagram, it is written only once because multiple arrows can be drawn from it. Therefore, the inputs for the given relation are 4, 0, -3, and 8.
If the relation is given as a vertical table, the inputs are the values in the first column.
2
List the Outputs
expand_more Write all the outputs in a second column. When the relation is given as a set of coordinates, the outputs are the second components of each pair.
{(4, 3),(0, -5),(-3, 3),(8, 0),(4, -5)}
The number - 5 is an output for two different pairs. In the mapping diagram, it is written only once because multiple arrows can be drawn to it. For the given relation, the outputs are 3, -5, and 0.
If the relation is given as a vertical table, the outputs are the values in the second column.
3
Connect Inputs and Outputs
expand_more Draw an arrow from each input to its corresponding output.
4
Look For Arrows With the Same Tail
expand_more Check if there are multiple arrows with the same starting input. In other words, identify inputs that have multiple outputs.
Here, the input 4 has two different outputs, 3 and -5. By definition, for a relation to be a function, every input must be assigned to exactly one output. As such, the given relation is not a function.
Method
Vertical Line Test
The vertical line test is a graphical method to determine whether a given relation is a function. For example, consider the following relations.
| Relation I | Relation II | Relation III |
|---|---|---|
| (y - 2)(y + 1) = x | y=x(x+1)(x-2) | |c|c|c|c|c|c|c|c| x & -2.25 & -1.75 & -1 & -1 & 0 & 1 & 2 y & -2 & 0 & 2 & -1 & 1 & -0.75 & 0.5 |
To determine whether the relations are functions, follow these two steps.
1
Draw the Relation on the Coordinate Plane
expand_more Draw the relation on the coordinate plane. The first two relations can be drawn using a graphing calculator or other mathematical software.
2
Draw a Vertical Line and Look at the Intersection Points
expand_more Draw vertical lines at different places through the coordinate plane. If one of the lines intersects the graph more than once, the relation is not a function. Conversely, if no vertical line cuts the graph more than once, the relation is a function.
Notice that l_3 cuts the first graph at two different points. The line m_2 also passes through two different points. This means that neither Relation I nor Relation III is a function. However, all of the vertical lines drawn over Relation II only intersect the graph one time at most. Because of this, Relation II is a function.
- Relation I is not a function. *
- Relation II is a function. ✓
- Relation III is not a function. *
Keep in mind that before stating whether a relation is a function, the vertical lines drawn have to cover the entire domain to ensure that no vertical line cuts the graph more than once.
Why
Intuition Behind the MethodIf a relation is not a function, it is because there are multiple y-values corresponding to the same x-value. Therefore, the graph of such relation would show at least two points directly above the other.
Points with the same x-value belong to the same vertical line.
This is why drawing a vertical line and moving it across the graph reveals if the graph is a function or not.
Note that when determining whether a relation is or is not a function, it must be assumed that the graph of a relation continues without any significant change beyond the boundaries of the coordinate plane. If this were not the case, it could never be determined from a graph whether a relation is a function.
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Example
Analyzing Relations Using Mapping Diagrams
Ignacio enjoyed displaying the relations between the ages and heights of Kriz's teammates so much that he wanted to go a step further and continue analyzing relations. He then asked Ramsha and Diego the ages and heights of their family members and wrote that data, including each player's values, in the following sets.
Note that (18,1.75) means that some member of Ramsha's family is 18 years old and 1.75 meters tall.
a What is the domain of R?
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{"type":"text","form":{"type":"list","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false,"ordermatters":false,"numinput":4,"listEditable":true,"hideNoSolution":true},"text":" "},"formTextBefore":"Range={","formTextAfter":"}","answer":{"text":["1.36","1.75","1.82","1.85"]}}
b What is the domain of D?
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{"type":"text","form":{"type":"list","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false,"ordermatters":false,"numinput":4,"listEditable":true,"hideNoSolution":true},"text":" "},"formTextBefore":"Range={","formTextAfter":"}","answer":{"text":["1.72","1.88","1.91","1.94"]}}
c Using mapping diagrams, determine which of the sets represents a function?
{"type":"choice","form":{"alts":["Only Set R","Only Set D","Both","None"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}
Hint
a The domain of a set of coordinate pairs is formed by all the coordinates written first, while the range is formed by all the coordinates written second. When writing sets, do not repeat elements.
b To find the domain, look at the first coordinates of each pair. Similarly, to find the range, look at the second coordinate of each pair. When writing sets, do not repeat elements.
c When represented as a mapping diagram, if only one arrow is drawn from each input, the relation is a function.
Solution
a Given a set of coordinate pairs, the domain includes all the first coordinates, and the range includes all the second coordinates.
By listing the numbers from least to greatest, and including each element only once, the domain and range of R are as follows. Domain ofR &= {12,18,23,43,47} Range ofR &= {1.36,1.75,1.82,1.85}
b As done in Part A, to find the domain of D, look at the first coordinate of each pair. To find the range, look at the second coordinate of each pair.
By listing the numbers from least to greatest, and including each element only once, the domain and range of D are as follows. Domain ofD &= {15,21,41} Range ofD &= {1.72,1.88,1.91,1.94}
c Start by drawing a mapping diagram representing the relation given by the sets R and D. The elements in the domains — the ages — will be grouped in one column, and the elements in the ranges — the heights — will be grouped in a second column.
Then, each age will be connected to the corresponding height using an arrow.
Remember that for mapping diagrams, a relation is a function if there is only one arrow drawn from each input to an output. In this case, the age is the input.
In the first mapping diagram, each age is assigned to only one height. Therefore, R represents a function. On the contrary, in the second mapping diagram, 41 is assigned to two different heights. In that case, set D cannot represent a function.
Only set R represents a function.
By analyzing the two data sets, some other conclusions can be drawn.
- Ramsha's guardians are of different ages and are the same height.
- Ramsha has two siblings.
- Diego's guardians are both 41 years old.
- Diego has one sibling.
Of course, based on the given information, it does not have to be the exact situation. The conclusions were done simply to show one way of interpreting the data recorded.
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Example
Analyzing Graphs Using the Vertical Line Test
Having completed some analysis for Kriz's volleyball team, Ignacio now has some free time. He is digging around his school's library computer. He finds some juicy information about a recently created virtual coin. There is a graph that shows the fluctuation in the coin's exchange rate during its first ten days after it was created.
a Find the domain and range of the first graph. Write the answer using compound inequalities.
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{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["y"],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"Range:","formTextAfter":null,"answer":{"text":["0.5\\leq y \\leq 8"]}}
b Find the domain and range of the second graph.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"Domain:","formTextAfter":null,"answer":{"text":["0.5\\leq x \\leq 8"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["y"],"constants":[]},"rendered":false},"text":" "},"formTextBefore":"Range:","formTextAfter":null,"answer":{"text":["0 \\leq y \\leq10"]}}
c Which of the two graphs represents a function?
{"type":"choice","form":{"alts":["Only the First Graph","Only the Second Graph","Both Graphs","None"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}
Hint
a The domain is all the x-values for which the graph is drawn. Similarly, the range is all the y-values for which the graph is drawn.
b The domain goes from the minimum to the maximum value along the x-axis for which the graph is drawn. Similarly, the range goes from the minimum to the maximum value along the y-axis for which the graph is drawn.
c Use the Vertical Line Test.
Solution
a To find the domain of the graph, travel along the graph making note of the x-values.
As seen, the graph is drawn for every value from 0 to 10, both ends included. Therefore, the domain is 0≤ x ≤ 10. To find the range, look at all the y-values for which the graph is drawn.
The range goes from 0.5 to 8, both endpoints included. Therefore, the range is 0.5≤ y ≤ 8.
b As in Part A, to find the domain of the graph, travel along the graph making note of the x-values.
From the above, the domain is 0.5≤ x ≤ 8. Next, look at the y-values.
The range goes from 0 to 10. It can be written as 0≤ y ≤ 10.
c To determine if the first graph represents a function, the Vertical Line Test can be applied. To do so, draw a vertical line, move it horizontally along the domain, focusing on the intersection point.
As seen, any vertical line never intersects the graph more than once. That means the first graph represents a function. Now, repeat the same procedure for the second graph.
Unlike the previous graph, here, there is at least one vertical line that intersects the graph more than once. This graph does not represent a function.
Only the first graph represents a function.
At first, Ignacio was glad that his mistake only caused the graph to change domains and ranges. He only realized that it was no longer a function when he applied the vertical line test. He better change it back before someone needs to use the database!
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Pop Quiz
Analyzing Different Relations
The following applet displays a relation either as a set of ordered pairs, a group of coordinate points in the coordinate plane, or a curve in the coordinate plane. For any given relation, determine whether it is a function.
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Closure
Restricting a Relation to Make It a Function
Consider a relation represented by the following graph. Note that any vertical line drawn between -4 and -2 intersects the graph more than once. Then, by the Vertical Line Test, the graph is not a function. However, by changing the relation's domain using x_1 and x_2, it is possible to convert the graph into a function.
As before, the initial graph is not a function, but it can be converted into a function by restricting either its domain or range. By moving x_1 and x_2, investigate whether restricting the domain will make the graph a function.
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Relations and Functions
Exercise 3.1
Consider the relation F whose graph is shown below. The domain is all the real numbers.
The relation can be converted to a function by restricting the domain. In which of the following domains is the relation a function?
{"type":"multichoice","form":{"alts":["D_1 = (-\u221e,-6.5]","D_2 = [-6,0]","D_3 = [-3,3]","D_4 = [0.5,4.5]","D_5 = [0.5,6]","D_6= [5,\u221e)","D_7 = [6.5,\u221e)"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":[0,3,6]}
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To determine whether the relation R becomes a function when its domain is restricted to the given intervals, we first restrict the graph, then we use the Vertical Line Test. We will consider each interval, one at a time.
Testing D_1= (-∞,-6.5]
Let's start by restricting the graph to the domain D_1 = (-∞,-6.5].
Now we draw vertical lines inside this frame.
As we can see, any vertical line drawn inside this domain intersects the graph only once. Therefore, when the relation is restricted to D_1= (-∞,-6.5], it becomes a function.
Testing D_2= [-6,0]
Next we restrict the graph to the domain D_2 = [-6,0].
Next, we draw vertical lines inside this frame.
As we can see, the vertical lines intersect the graph more than once. This implies that, in D_2= [-6,0], the relation is not a function.
Testing D_3= [-3,3]
Let's restrict the graph to the domain D_3= [-3,3].
Now, we draw vertical lines inside this frame.
We can see that some lines intersect the graph once and others intersect it more than once. Therefore, the relation is not a function in D_3 = [-3,3].
Testing D_4= [0.5,4.5]
Next we check the interval of D_4= [0.5,4.5].
Let's draw some vertical lines inside this frame.
Any vertical line drawn between 0.5 and 4.5 intersects the graph only once. Consequently, the relation R is a function when its domain is restricted to D_4= [0.5,4.5].
Testing D_5= [0.5,6]
Let's continue with D_5= [0.5,6].
Now, we will draw some vertical lines inside the frame.
We can see that some lines intersect the graph once and others intersect it more than once. Then, the relation is not a function in D_5 = [0.5,6].
Testing D_6= [5,∞)
We are almost done. Let's check now D_6= [5,∞).
As before, let's draw some vertical lines inside this frame.
As in the previous part, some lines intersect the graph once and others intersect it more than once. Consequently, the relation is not a function in D_6 = [5,∞).
Testing D_7= [6.5,∞)
Finally, let's try D_7= [6.5,∞).
For the last time, let's draw some vertical lines inside this frame.
Any vertical line drawn to the right of 6.5 intersects the graph only once. As such, the relation F is a function when its domain is D_7= [6.5,∞).
Conclusion
After checking each interval, we have reached the following conclusions.
| Domain | Is F a function? |
|---|---|
| D_1 = (-∞,-6.5] | ✓ |
| D_2= [-6,0] | * |
| D_3= [-3,3] | * |
| D_4= [0.5,4.5] | ✓ |
| D_5= [0.5,6] | * |
| D_6= [5,∞) | * |
| D_7= [6.5,∞) | ✓ |
In general, the biggest domain in which the relation F is a function is (-∞,-6)⋃(0,5)⋃(6,∞).
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