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Algebra 1 View details

1. Relations and Functions

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eCourses /

Algebra 1 /

Chapter 3

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.

Lesson Settings & Tools

	11 Theory slides
	11 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

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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.

Two graphs, one is a function and the other is a not

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.

This mapping diagram shows the relation defined by the set

{(- 2, - 8),

(- 1, 11),

(0, 5),

(2, - 8)} .

Note that

- 8

is the output of two different inputs,

- 2

and

2 .

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.

Tiffaniqua, 18 yo, 1.70m; Kriz, 20 yo, 1.96m; Ramsha, 23 yo, 1.85m; Jordan, 20 yo, 1.75m; Diego, 21 yo, 1.91m; LaShay, 19 yo, 1.87m

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.

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.

Note that every function is a relation, but not every relation is a function. In the following applet, three different relations are analyzed to determine whether they are functions.

Three different mapping diagrams. The second one is not a function because one input has two outputs.

In Relation III, although one of the outputs corresponds to two different inputs, it is still a function because each input has exactly one output. Depending on how a relation is represented, there are different methods to determine whether or not it is a function.

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.

f (x) g (x) h (x) = 3 x = x = \frac{1}{x}

Their domains can be written by analyzing the definition of each function.

Function	Analysis	Domain
$f (x) = 3 x$	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, $x \geq 0$
$h (x) = \frac{1}{x}$	Division by zero is undefined.	All real numbers except $0$ — that is, $x \neq = 0$

Depending on how a function is represented, its domain can be determined by using different methods.

Graph in the coordinate plane, table of values, set of coordinate pairs, and mapping diagram

The domain of a function also depends on what the function describes. For example, let

p (x) = 2 x

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.

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
$f (x) = 2 x$	All integers
$g (x) = x^{2}$	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
$f (x) = 2 x$	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, $y \geq 0$
$h (x) = 4$	All real numbers	The function takes any real number input and sends it to $4 .$	Only the number $4$ — that is, the range is ${4}$

Depending on how a function is represented, its range can be determined using different methods.

If two different inputs have the same output, it is not necessary to repeat the output when writing the range.

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!

Machine simulating the function f(x)=x+2. Every number that enters in the machine is increased by 2.

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.

A laptop with the following code written: if color=0, then print in black; if color=1, then print in blue; if color=2, then print in green; if color=0, the print in red.

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.

List the Inputs

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 .

The input numbers 4, 0, -3, and 8 listed in a column.

If the relation is given as a vertical table, the inputs are the values in the first column.

List the Outputs

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 .

The input numbers 4, 0, -3, and 8 listed in a left column, and the corresponding output numbers 3, -5, and 0 listed in a right column.

If the relation is given as a vertical table, the outputs are the values in the second column.

Connect Inputs and Outputs

Draw an arrow from each input to its corresponding output.

The input numbers 4, 0, -3, and 8 listed in a left column, and the corresponding output numbers 3, -5, and 0 listed in a right column. Pairs of inputs and corresponding outputs are shown with arrows: (4,3), (4,-5), (0,-5), (-3,3), and (8,0).

Look For Arrows With the Same Tail

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
$(y - 2) (y + 1) = x$	$y = x (x + 1) (x - 2)$	$x y - 2.25 - 2 - 1.75 0 - 1 2 - 1 - 1 01 1 - 0.75 2 0.5$

To determine whether the relations are functions, follow these two steps.

Draw the Relation on the Coordinate Plane

Draw the relation on the coordinate plane. The first two relations can be drawn using a graphing calculator or other mathematical software.

Graph of the three relations: (y-2)(y+1)=x; y=x(x+1)(x-2), and x (-2.25,-2),(-1.75,0),(-1,2),(-1,-1),(0,1),(1,-0.75), and (2,0.5)

Draw a Vertical Line and Look at the Intersection Points

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.

Different vertical lines drawn along each relation

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. $\times$
Relation II is a function. $✓$
Relation III is not a function. $\times$

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 Method

If 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.

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.

R = {(18,1.75),(23,1.85),(12,1.36),(43,1.82),(47,1.82)}, D = {(15,1.72),(21,1.91),(41,1.88),(41,1.94)}

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 ?

What is the range of

R ?

b What is the domain of

D ?

What is the range of

D ?

c Using mapping diagrams, determine which of the sets represents a function?

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 of R Range 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.

By listing the numbers from least to greatest, and including each element only once, the domain and range of

D

are as follows.

Domain of D Range 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.

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.

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!

a Find the domain and range of the first graph. Write the answer using compound inequalities.

b Find the domain and range of the second graph.

c Which of the two graphs represents a function?

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

10,

both ends included. Therefore, the domain is

0 \leq x \leq 10 .

To find the range, look at all the

y -

values for which the graph is drawn.

The range goes from

0.5

8,

both endpoints included. Therefore, the range is

0.5 \leq y \leq 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 \leq x \leq 8 .

Next, look at the

y -

values.

The range goes from

0

10 .

It can be written as

0 \leq y \leq 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!

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

x_{1}

and

x_{2},

it is possible to convert the graph into a function.

Graph that is not a function but the domain can be restricted by moving x1 and x2

Now consider the graph of the relation

y = \pm 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

x_{1}

and

x_{2},

investigate whether restricting the domain will make the graph a function.

Graph of the function y= +- sqrt(x) where the domain can be restricted by moving x1 and x2

As can be seen, no matter where

x_{1}

and

x_{2}

are placed, the resulting graph cannot be made into a function. Not to give up too soon, try moving

y_{1}

and

y_{2},

to restrict the range.

Graph of the function y= +- sqrt(x) where the range can be restricted by moving y1 and y2

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.

\underline{Relation} y = \pm x y = x y = - x \underline{Function ?} \times ✓ ✓

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.

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Level 3

Relations and Functions

Exercises

Consider the following relation given as a mapping diagram.

A relation from the set [9,0,-10,-1] to the set [-4,2,7]

What is the domain of the relation?

What is the range of the relation?

When a relation is represented as a mapping diagram, the domain is the set formed by all the elements inside the input set. Let's point out these elements in the given diagram.

Note that elements in a set are usually, though not necessarily, written in order from least to greatest. This means that the elements of the domain of the given relation is the set {-10,-1,0,9}.

When a relation is represented as a mapping diagram, the range is the set formed by all the elements inside the output set. Let's point out these elements in the given diagram.

Although -4 is the output of two different inputs, we do not need to write -4 twice when writing the range. This is because elements of a set are written only once. Then, the range of the given relation, written from least to greatest, is the set {-4,2,7}.

Consider the following relation given as a mapping diagram.

A relation from the set [4.5,sqrt(2),-1,7] to the set [1,7,-4,-6.5,2]

What is the domain of the relation?

What is the range of the relation?

When a relation is represented as a mapping diagram, the domain is the set formed by all the elements inside the input set — the set from which the arrows start. Let's point out these elements in the given diagram.

As we can see, the domain of the given relation is the set {4,sqrt(2),-1,7}. Domain = {4,sqrt(2),-1,7} Usually, when writing the domain of a relation, it is convenient to order the elements from least to greatest so that we can easily identify the minimum and maximum values if needed. In our case, the domain of the relation, written from least to greatest, is the set {-1,sqrt(2),4.5,7}.

When a relation is represented as a mapping diagram, the range is the set formed by all the elements inside the output set — the set of elements that the arrows point to. Let's point out these elements in the given diagram.

From this, the range of the given relation, written from least to greatest, is the set {-6.5,-4,1,2,7}.

Consider the following graph of a relation.

A graph in the coordinate plane. f(x)=(x+9)*(x+3)*x*(x-5)*(x-6)/1000-2.2 from x=-9 to x=8.

What is the domain of the relation? Write the answer as a compound inequality.

What is the range of the relation? Write the answer as a compound inequality.

To find the domain of a graph, we look at all the x-values for which the graph is drawn. One way to do this is by going through the graph from left to right and noting the starting and ending x-values.

The starting x-value is -9 and the ending x-value is 8. Also, we can see that all the numbers in between are included in the graph. In other words, the graph is drawn for every value between -9 and 8. This implies that the domain of the given relation is -9 ≤ x ≤ 8.

To find the range of a graph, we look at all the y-values for which the graph is drawn. We will go through the graph in a similar way to what we did in the previous part, but this time we pay attention to the minimum and maximum y-values.

The minimum y-value is -3 and the maximum y-value is 7. In addition, all the numbers in between are included in the graph. That is, the graph is drawn for every y-value between -3 and 7. We can conclude that the range of the given relation is -3 ≤ y ≤ 7.

Consider the following graph of a relation.

The graph of a curve that is not a function (the right part of an ellipse and one portion of the sine function but rotated 90 degrees)

What is the domain of the relation? Write the answer as a compound inequality.

What is the range of the relation? Write the answer as a compound inequality.

To find the domain of a graph, we will look at all the x-values for which the graph is drawn. First, let's identify the minimum and maximum x-values for which the graph is drawn.

The minimum value is -5 and the maximum value is 9. Additionally, the graph is drawn for every number between these two values.

We can conclude that the domain of the given relation is -5 ≤ x ≤ 9.

To find the range of a graph, we look at all the y-values for which the graph is drawn. In a similar way to what we did in the previous part, we start by identifying the minimum and maximum y-values for which the graph is drawn.

We can see that the minimum value is -2 and the maximum value is 8. Also, the graph is drawn for every number between these two values.

Then, we can say that the range of the given relation is -2 ≤ y ≤ 8.

The following table represents the relation between two variables $x$ and $y,$ where $y$ depends on $x .$

$x$	$y$
$2$	$3$
$- 1$	$14$
$6$	$3$
$0$	$- 2$
$- 3$	$0$

What is the domain of the relation?

What is the range of the relation?

When a relation is given as a table of values, we first identify the independent and dependent variables. In our case, we are told that y depends on x, so we know that y is the dependent variable. Independent Variable &→ x Dependent Variable &→ y To find the domain from a table of values, we look at the row or column corresponding to the independent variable. In this case, we look at the column corresponding to the x-values.

x	y
2	3
-1	14
6	3
0	-2
-3	0

The domain of the given relation, written from least to greatest, is the set {-3,-1,0,2,6}.

From the previous part, we already know that x is the independent variable and y is the dependent variable. To find the range, we look at the row or column corresponding to the dependent variable. Let's look at the column corresponding to the y-values.

x	y
2	3
-1	14
6	3
0	-2
-3	0

Notice that 3 appears twice. However, when writing the range, it is enough to write the value only once. Therefore, we can say that the range of the given relation, written from least to greatest, is the set {-2,0,3,14}.

Consider a relation given as the following set of ordered pairs.

What is the domain of the relation?

What is the range of the relation?

The domain of a relation represented by a set of ordered pairs is formed by the first coordinates of the ordered pairs. Therefore, let's start by identifying the corresponding coordinates in the given pairs.

Notice that 3 appears twice in the set. However, when writing a set, it is enough to write each element once. Therefore, the domain of the given relation, written from least to greatest, is the set {-5,0,2,3,9}.

When a relation is given a set of ordered pairs, the range is represented by the second coordinates of all of the ordered pairs. Thus, we will start by pointing out the second coordinates.

As in the previous part, we can see that there are some repeated elements — namely, 0 and -2. However, when writing the range, we will write them only once, without repetition. Therefore, the range of the given relation, written from least to greatest, is the set {-2,0,1,6}.

Relations and Functions

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