THE LANGUAGE OF RELATIONS AND FUNCTIONS || MATHEMATICS IN THE MODERN WORLD

Summary

This educational video explores the concept of relations and functions in mathematics. It explains how relations link values from a domain to a range, using ordered pairs to illustrate. The video clarifies that a function is a type of relation where each domain value corresponds to exactly one range value. It further discusses how to determine if a relation is a function using methods like the vertical line test and mapping diagrams. Practical examples and equations are provided to demonstrate function evaluation. The video is an informative guide for learners seeking to understand the fundamentals of relations and functions.

Takeaways

• 📚 The script discusses the concept of 'relation' in mathematics, explaining how it is a rule that associates values from one set (domain) to another (range).
• 🔗 It illustrates the idea of a relation as a set of ordered pairs, where each pair represents a connection between elements of two different sets.
• 🧑‍🏫 The video uses the analogy of a machine that takes inputs (from the domain) and produces outputs (from the range) according to a certain rule.
• 📈 The concept of a function is introduced as a specific type of relation where each input is associated with exactly one output.
• 🔍 The script differentiates between relations and functions by explaining that a function cannot have two different outputs for the same input.
• 📝 Examples of ordered pairs are given to demonstrate which pairs are part of a relation and which are not, based on the rule provided.
• ❓ The video poses questions to engage viewers in determining whether certain pairs are part of a given relation and to identify the domain and range of that relation.
• 📊 It explains how to determine if a relation is a function by using the vertical line test on graphs and by examining if each x-value has a unique corresponding y-value.
• 📚 The script also covers different ways to represent functions, such as through tables, ordered pairs, graphs, and equations.
• 📉 The vertical line test is described as a method to determine if a graph represents a function, where a vertical line should intersect the graph at most once.
• 📝 The process of evaluating a function for a given input value is demonstrated with examples, showing step-by-step calculations.

Q & A

What is the language of relation in mathematics?

The language of relation in mathematics refers to the way values from one set, known as the domain, are related to a second set of values, known as the range, through a rule that defines the relationship between elements of these sets.

What is a relation in the context of sets and ordered pairs?

A relation is a set of ordered pairs (x, y) where each pair represents a relationship between elements x from the domain and y from the range according to a certain rule.

How is a relation represented in terms of a machine and its inputs and outputs?

A relation can be thought of as a machine where the elements of the domain are inputs, and the machine applies a rule to these inputs to generate one or more outputs, which are the elements of the range.

What is an example of an ordered pair in a relation?

An example of an ordered pair in a relation could be (1, 2), which means that the element 1 from the domain is related to the element 2 from the range according to the defined rule of the relation.

What is a function in mathematics?

A function is a special type of relation where each element in the domain is related to exactly one element in the range. In other words, no two ordered pairs in a function have the same first element but different second elements.

How can you determine if a relation is a function using the vertical line test?

The vertical line test states that a graph represents a function if and only if each vertical line intersects the graph at most once. If any vertical line intersects the graph more than once, the relation is not a function.

What is the difference between a relation and a function in terms of the number of outputs per input?

In a relation, an input can correspond to one or more outputs, whereas in a function, each input corresponds to exactly one output.

How can you represent a function in different ways?

A function can be represented in various ways, including a table of values, ordered pairs, a graph, a mapping diagram, or an equation.

What is the domain of a relation or function?

The domain of a relation or function is the set of all possible input values (x-values) for the relation or function.

What is the range of a relation or function?

The range of a relation or function is the set of all possible output values (y-values) that result from applying the rule to the domain.

How can you evaluate a function given an input value?

To evaluate a function for a given input value, you substitute the input value into the function's equation and perform the necessary calculations to find the corresponding output value.

Outlines

00:00

🔗 Understanding Relations and Functions

This paragraph introduces the concept of relations and functions in mathematics, which are prevalent in everyday life. It explains that a relation is a rule that associates elements from one set (domain) to another (range), and it can be represented as a set of ordered pairs. The video aims to answer questions about specific ordered pairs and their relation to a given rule, identifying which pairs satisfy the condition of the relation. It also discusses the domain and range of a relation, using an example with sets A and B to illustrate how to determine if certain pairs are part of the relation.

05:01

📚 Exploring the Characteristics of Functions

The second paragraph delves into the specifics of functions, which are a type of relation where each element in the domain is associated with exactly one element in the range. It contrasts functions with general relations and uses examples to show which sets of ordered pairs represent functions and which do not. The paragraph also covers different ways to represent functions, such as tables of values, ordered pairs, graphs, and equations, and it explains the concept of one-to-one and onto functions using mapping diagrams.

10:03

📉 The Vertical Line Test for Functions

This part of the script focuses on the vertical line test, a graphical method to determine whether a curve represents a function. It states that a curve is a graph of a function if and only if every vertical line intersects the curve at most once. The paragraph provides examples of graphs that pass and fail the vertical line test, including linear and quadratic functions, and explains how to identify functions using equations, with examples of both valid functions and those that are not.

15:07

📝 Evaluating Functions with Given Inputs

The final paragraph demonstrates how to evaluate functions with specific inputs. It walks through the process of substituting values into function equations to find the corresponding outputs. Two examples are given: the first evaluates a quadratic function when x equals two, and the second evaluates a linear function when x equals three. The paragraph concludes with a summary of the results and a thank you note to the viewers, encouraging them to like, subscribe, and stay updated for more educational content.

Keywords

💡Relation

A 'relation' in the context of this video refers to a mathematical concept where a set of values (domain) is associated with another set of values (range) according to a specific rule. It is central to the video's theme as it sets the foundation for understanding the relationships between different elements in mathematics. For example, the video discusses how relations can be represented as ordered pairs, such as (1,2), (2,4), etc., to illustrate the connection between elements in the domain and the range.

💡Domain

The 'domain' is defined as the set of possible input values for a relation or function. It is a fundamental concept in the video, as it represents the starting point for any relational mapping. The script uses the domain to explain how each element within it can be related to elements in the range, such as in the relation r where the domain consists of the numbers 1, 2, and 3.

💡Range

'Range' is the set of possible output values that result from applying the rule of a relation to the domain. It is integral to the video's explanation of how relations work, as it shows the possible outcomes of the relational process. The video script mentions the range when discussing the relation r, indicating the set of values that each element in the domain can be related to, such as 2, 4, 6, 8, and 10.

💡Ordered Pair

An 'ordered pair' is a pair of elements where the order is significant, typically used to represent elements of a relation. The video uses ordered pairs to illustrate the concept of relations, such as (1,2) indicating that the element '1' from the domain is related to the element '2' in the range according to the defined rule.

💡Function

A 'function' is a special type of relation where each element of the domain is associated with exactly one element in the range. The video emphasizes the uniqueness of this relationship, contrasting it with general relations. For instance, it explains that in a function, there cannot be two ordered pairs with the same first element but different second elements.

💡Mapping Diagram

A 'mapping diagram' is a visual representation used to illustrate the relationship between the domain and range in a function. The video uses mapping diagrams to show how each input value from the domain corresponds to a unique output value in the range, helping to clarify the concept of a function and its properties.

💡Vertical Line Test

The 'vertical line test' is a graphical method to determine if a curve represents a function. If any vertical line intersects the curve more than once, it does not represent a function. The video script refers to this test when discussing the graphical representation of functions, emphasizing that only curves that pass the vertical line test can be considered functions.

💡Quadratic Function

A 'quadratic function' is a type of function represented by an equation of the form y = ax^2 + bx + c, where a, b, and c are constants, and 'a' is not zero. The video mentions quadratic functions as examples of functions that can be represented graphically as parabolas, which are curves opening upwards or downwards.

💡Linear Function

'Linear function' refers to a function represented by a straight line when graphed. The video script includes linear functions as examples of functions that pass the vertical line test, indicating their simple and direct relationship between the domain and range.

💡Evaluate

To 'evaluate' a function means to find the output value for a given input value. The video demonstrates how to evaluate functions by substituting specific values of x into the function's equation and simplifying to find the corresponding y value, as shown in the examples provided for functions f(x) and q(x).

Highlights

The video discusses the concept of relations and functions, which are fundamental in mathematics and have practical applications in daily life.

Relations are defined as a rule that connects values from a domain to a range, illustrating the interconnectedness in various contexts such as family, education, and business.

A relation is represented as a set of ordered pairs, demonstrating how each element in the domain corresponds to an element in the range.

The video explains that the elements of the domain can be thought of as inputs to a machine that generates outputs based on a given rule.

An example of a relation is provided, showing specific ordered pairs and their corresponding outputs, emphasizing the relationship between inputs and outputs.

The video introduces the concept of the domain and range of a relation, explaining how they define the possible inputs and outputs.

A method to determine if a given pair is part of a relation is presented, using the criterion that the difference between x and y divided by 2 must be an integer.

The video poses questions to engage viewers in identifying which pairs are part of the relation and which are not, based on the given rule.

Functions are introduced as a type of relation where each element in the domain is related to exactly one element in the range.

The video explains that functions can be represented in various ways, including tables of values, ordered pairs, graphs, and equations.

Different relations are evaluated to determine whether they are functions, using the criteria that no two pairs can have the same x-value with different y-values.

The concept of mapping diagrams is introduced as a visual tool to represent functions, showing a unique output for each input.

The video uses the vertical line test to determine if a graph represents a function, explaining that a function's graph will intersect a vertical line at most once.

Examples of linear and quadratic functions are given, demonstrating how they can be identified as functions using the vertical line test.

The video explains how to evaluate a function given an input value, showing step-by-step calculations for specific functions.

The importance of understanding the difference between relations and functions is emphasized, highlighting their unique properties and applications.

The video concludes with a summary of the key points, encouraging viewers to practice identifying relations and functions in various mathematical contexts.

Transcripts

in this video we are going to discuss

the language of relation

and functions

relations abound in daily life people

are related to each other in many ways

as parents and children teachers and

students

employers and employees and many others

in business things that are both are

related to their costs and the amount

paid is related to the number of

things both when you say relation

it is a rule that relates values from a

set of values

we call that as a domain to a second set

of values and we call that

as a range the elements of the domain

can be

imagined as input to a machine

that applies a rule to these inputs to

generate

one or more outputs

a relation is also a set of ordered pair

x and y for example

a relation r have a set of ordered pair

one comma two two comma four three comma

six four comma eight

five comma ten so young one

two three four and five eight in a tower

nothing domain

young two four six eight and ten eighty

nine

range

okay a relation is a set as a subset so

for example

let's set a is equal to one and two and

set

b is equal to one two three and define a

relation r

from a to b as follows so given the

statement

that your x and y is an element of the

product of

set a and b so x and y is an element of

relation r

it means that so by the by this

statement

x minus y over 2 is an

integer so we're going to answer the

following questions

state explicitly which ordered pairs and

a times b and which are in relation

are is 1 related to 3

is 2 related to 3 is 2 related to 2

what are the domain and range of

relation r

okay let's answer number one question so

first

we need to get the product of set a

and b so to get the product so that is

one one say to yan one one

one two one three

two one two two

and two three so all

of this no this set of ordered pairs so

is i

saying nothing is a substitute on the

given statement so

long and making any mean silent relation

are capac

one in one is an element of relation r

y so apache and the given statement

integer and sagot not n so pakistan be

nothing integers

so y and i included john young negative

numbers zero

and positive numbers since the

output is zero and zero is an integer so

therefore

one and one is an element of relation

r next one two one two is not

an element of relation r y capacitor

nothing negative one half fraction

so this is not and integer

one tree is an element of relation r why

so synaptic nothing on the given

statement

not a negative one a negative one is

an integer so e big sub n one and 3 as

an element of

relation r next 2

and 1 is not an element of relation r y

so nothing capacitive

given statement so not in one hub and

one half is not an integer

another 2n2 is an element of relation ry

so not in change 0 so 0 is

again is an integer

next 2 3 is not an element of relation r

because negative one half is not an

integer next

so therefore yumanga element

na in relation r so that is

one one one three and

two two

number two question is one related to

three

is two related to three is two related

to two so

given that in kanina so do not impact

question number two so is 1 related to 3

yes because 1 and 3 is an element of

relation r and then 2 is related to 3

the answer is no because

2 and 3 is not an element

of relation

because negative one happy output not in

gen

and negative one half is not an integer

next is two related to two the answer is

yes because 2 is an element of

relation r question number 3 what are

the domain and range of

r so in the given problem set a

is our x values right and set b is our y

values so therefore the domain for

relation r

is one and two and the range for

set for this given so that is our b

that is one two and three

function when you say function it is a

relation

where each element in the given

in the domain is related to only one

value in the range by some rule

next it is the element of the domain can

be

imagined as input to a machine that

applies a rule so that each input

corresponds to only one output so kanina

don't define that in c relation so you

output

it's pretty much in one or more output

function only one

output another

a function is a set of ordered pairs

x and y such that no

two ordered pairs have the same x value

but different y values

ordered pairs the middle same x values

and doublet meron silang different

y values or makkah bayong output the

pattern

okay function can be represented in

different ways so

again a table of values

ordered pairs graph

and equation so my kitanathan so we can

represent function in this

four different ways so which of the

following relation are found

which of the following relation our

function first relation

f is equal to one comma two two comma

two

three comma five four comma pi so

function by end

yes it is a function bucket x

value

so in relation g we have one comma three

one comma four

two comma five two comma six and three

comma seven

so function bashar this is not a

function bucket

x value

okay next

in relation h we have 1 comma 3 2 comma

6 3 comma

9 up to n comma 3 n so

in this given function or not a function

of course that is a function so

nothing in value

so therefore relation h is a function

okay also function can be represented

using mapping diagrams

so in the first

mapping diagram so thingy nathan

our x values a meron unique

output e b sub n one two one sila

so say some x values melon is some y

value

so my own that is a function so that and

then i

one to one using mapping diagram that is

a function

next okay maritime

5 7 the input or x value name

is output long that is one at meron

time six eight and nine name is output

log then

therefore antonio not in detail is

mainly to one capac may need to one that

is also a function okay

one is to one or may need to one that

is a function okay in the third mapping

diagram many time

seven corresponds at the one

y value not n e big sub n

since uh c7 metal eleven

and thirteen so antagonized

one too many

that is a function may need to one that

is also a function

but if there is one too many that is not

a function

the vertical line test this is a graph

represent a function

if and only if each vertical line

intersects the graph

at most once so guinea gamete is a

vertical line test

but i'm identifying nothing

but a function or not a function so

capacitor hit them in a vertical line

at some point

points a graph and i not a function

so for example in this graph

okay so kappa guinea into the vertical

line test guide some parting and graph

john

is

is

so the report is not a function

this one this is also not a function

using the vertical line so the one point

no graph nothing

okay using the equation we can identify

the given equation if that is a function

or not

bucket okay panetto this is an example

of linear functions

vertical i know uh straight line

tama so uh nothing on vertical line test

organization

linear function which is straight line

quadratic function quadratic function

linear function quadratic function is

a function also packet and graph

depending

quadratic function is a parabola

y okay

next okay y is equal to square root of x

plus one this is also a

function next y is equal to two x plus

one over x minus one

also a function okay

uh

that is not a function next

in evaluating a function so for example

q of x is equal to x squared minus two x

plus two when your x is equal to two so

on gagawin

and then simplify two squared is four

negative two times two that is negative

four plus two

four minus four that is zero plus two e

b sub union q

of two is equal to two another

we have f of x is equal to two x plus

one when your x is

three x minus one so on an e big sub

nito papalitan

x nang three x minus one

so uh f of three x minus one is equal to

two x plus one so papadi turned out and

excellent three x minus one

and then distribute nothing in two

sloped parenthesis so

two times three x that is six x two

times negative one that is negative two

plus one so uh and then simplify

it combine similar terms since

uh my one variable to six x naught so

just copy

the negative two plus one the answer is

negative one

thank you for watching this video i hope

you learned something

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