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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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| | 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.
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.
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.
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)
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.
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.
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)
| 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, or inputs, for which the function is defined. For example, consider the following functions.
Depending on how a function is represented, its domain can be determined by using different methods.
The domain of a function also depends on what the function describes. For example, let p(x)=2x be a function representing the price of x apples at a market. Although the function is defined for all real numbers, it does not make sense to find the price of a negative number of apples or a fraction of an apple. Here, the domain of p(x) is all non-negative integers.
f(x)g(x)h(x)=3x=x=x1
Their domains can be written by analyzing the definition of each function. | Function | Analysis | Domain |
|---|---|---|
| f(x)=3x | Multiplication by 3 is defined for all real numbers. | All real numbers |
| g(x)=x | Square roots are not defined for negative numbers. | All non-negative numbers — that is, x≥0 |
| h(x)=x1 | Division by zero is undefined. | All real numbers except 0 — that is, x=0 |
Concept
Range
The range of a function is the set of all y-values, or outputs, of the function. The range of a function 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)=x2 | All real numbers |
| h(x)=4 | All real numbers |
By analyzing the definition of each function along with the given domains, the ranges can be determined.
| 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)=x2 | 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} |
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!
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) | xy-2.25-2-1.750-12-1-1011-0.7520.5
|
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 ℓ3 cuts the first graph at two different points. The line m2 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.
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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b What is the domain of D?
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c Using mapping diagrams, determine which of the sets represents a function?
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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.
Domain of RRange of R={12,18,23,43,47}={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.
Domain of DRange of D={15,21,41}={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.
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.
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.
Ignacio accidentally deletes some part of the database code, and whoa! The graph switched the axes labels. Take a look at the shape of the graph. It changed. Ignacio figures he can use what he has learned recently and make sense of all this!
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.
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.
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.
a Find the domain and range of the first graph. Write the answer using compound inequalities.
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b Find the domain and range of the second graph.
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c Which of the two graphs represents a function?
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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.
b As in Part A, to find the domain of the graph, travel along the graph making note of the x-values.
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.
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!
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.
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 x1 and x2, it is possible to convert the graph into a function.
Now consider the graph of the relation y=±x, where the domain is all non-negative numbers and the range is all real numbers. 
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 x1 and x2, investigate whether restricting the domain will make the graph a function.
As can be seen, no matter where x1 and x2 are placed, the resulting graph cannot be made into a function. Not to give up too soon, try moving y1 and y2, to restrict the range.
There it is! In this case, the graph becomes a function when the range is restricted to a set containing either only non-negative or only non-positive numbers.
Relationy=±xy=xy=-xFunction?×✓✓
In conclusion, when given a relation that is not a function, it might be useful to check whether restricting its domain or range can make the relation become a function.
Relations and Functions
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