Relations and Functions | General Mathematics | Grade 11

00:14:28
https://www.youtube.com/watch?v=YU43Jvr3fs8

Summary

TLDRIn this educational video, the instructor discusses the concepts of relations and functions in mathematics. A relation is defined as a set of ordered pairs, where the domain consists of all the x components and the range consists of all the y components. The video explains how each member of the domain can be paired with only one member of the range to qualify as a function. The instructor also introduces the vertical line test, a graphical method to identify functions. The test suggests that if any vertical line crosses a graph at more than one point, it is not considered a function. Mapping diagrams are shown as a means of depicting functions by connecting domain elements to range elements through arrows. The instructor provides several examples, illustrating how to identify whether certain relations are functions or not, both algebraically and graphically. The video concludes with a brief summary and invites viewers to ask further questions in the comments.

Takeaways

  • 📘 A relation is a set of ordered pairs, having a domain and range.
  • 🔤 The domain is the set of all x components, or inputs, from the ordered pairs.
  • 🟢 The range is the set of all y components, or outputs, from the ordered pairs.
  • ✔️ A function is a relation where each domain element maps to exactly one range element.
  • 📏 The vertical line test helps determine if a graph is a function if only touches at one point.
  • 🔗 Mapping diagrams visually show how domains and ranges connect via functions.
  • ⚠️ A graph or relation where x-values have multiple y-values is not a function.
  • 🖋️ Examples in the video show how to determine functions from relations.
  • 📈 Functions can be graphically represented in the Cartesian plane.
  • 🧠 Understanding domain and range is crucial in differentiating mathematical functions.

Timeline

  • 00:00:00 - 00:05:00

    The video begins with an introduction to the concept of relations and functions in mathematics. A relation is defined as a set of ordered pairs, with the domain being all the x-components and the range being all the y-components of these pairs. An example is provided with sets of ordered pairs, and the domain and range are identified. Furthermore, a relation is explained as a rule that assigns values from the domain to values in the range. The video differentiates between relations and functions, explaining that a function pairs each member of the domain with exactly one member of the range, which means no two ordered pairs have the same x-value but different y-values. Two examples are given to illustrate this concept, demonstrating how to identify both a function and a non-function relation.

  • 00:05:00 - 00:14:28

    The video continues by discussing ways to represent functions, such as using mapping diagrams where domain elements are mapped to range elements using arrows. It shows examples of mapping diagrams and evaluates their functions. Furthermore, it introduces the vertical line test, a method to graphically determine if a relation is a function: if a vertical line intersects a graph at exactly one point, the graph represents a function. Examples of graphs are given to illustrate the use of this test. The video concludes with reminders to subscribe and invitations for viewers to ask questions in the comments section. This summary captures the logical structure and key points discussed regarding relations, functions, and how to visually determine their properties.

Mind Map

Video Q&A

  • What is a relation in mathematics?

    A relation is any set of ordered pairs, where the set of all x components is called the domain and the set of all y components is called the range.

  • What is the domain of a relation?

    The domain is the set of all x components (or input values) in the ordered pairs of a relation.

  • What is the range of a relation?

    The range is the set of all y components (or output values) in the ordered pairs of a relation.

  • How can you determine if a relation is a function?

    A relation is a function if each x-value (domain element) is paired with exactly one y-value (range element). You can also use the vertical line test on a graph.

  • What is the vertical line test?

    The vertical line test is a method to determine if a graph represents a function, by checking if any vertical line crosses the graph at more than one point.

  • What is the purpose of mapping diagrams in functions?

    Mapping diagrams show how elements of the domain are paired with elements of the range, using arrows to represent the connections.

  • Can a relation have the same x-value with different y-values to be a function?

    No, a function must have each x-value uniquely paired with one y-value, otherwise it is not a function.

  • What are examples of ordered pairs given in the video?

    Examples include (1,3), (2,4), (5,7), (6,8) for domains and ranges discussions.

  • What graphical element is used to represent functions in the video?

    Functions are represented using graphs in the Cartesian plane and mapping diagrams.

  • What are some examples of functions identified in the video?

    Examples include linear functions, shown through straight-line graphs where the vertical line test confirms them as functions.

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  • 00:00:01
    [Music]
  • 00:00:14
    hi class welcome back to our channel for
  • 00:00:17
    this video discussion
  • 00:00:19
    and about ebooks
  • 00:00:20
    some functions and
  • 00:00:22
    relations
  • 00:00:24
    okay so define when nothing young
  • 00:00:27
    relation
  • 00:00:28
    a relation is any set of ordered pairs
  • 00:00:31
    the set of all the x components
  • 00:00:35
    of the ordered pairs is called the
  • 00:00:37
    domain
  • 00:00:38
    of the relation
  • 00:00:39
    and the set of all the y components is
  • 00:00:43
    called the range
  • 00:00:45
    okay so if it's a bn a relation is a
  • 00:00:48
    rule
  • 00:00:49
    that relates values from a set of values
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    called the domain
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    okay to a second set of values called
  • 00:00:58
    the range
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    so whether nothing imagine domain
  • 00:01:03
    is your adding input
  • 00:01:05
    machine
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    while syringe is
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    so
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    let's give the domain and range of the
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    following relation for number one
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    we have one three
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    two four
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    five seven and six comma eight
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    x components of the ordered pairs
  • 00:01:37
    okay
  • 00:01:38
    so on only on we have one
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    two
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    five
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    and six
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    nahua
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    while young range number
  • 00:01:49
    is the set of all y components so you
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    know
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    we have seven
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    then the sixth meron eight
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    guys
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    three four seven and eight
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    so next number two
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    so
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    begin adding in domain and range
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    again your adding domain is the set of
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    all x components so you know
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    numbers we have negative two negative
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    one
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    then multiply negative two so since mean
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    negative two naught
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    that is our y components so we have four
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    one
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    zero
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    then five
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    and last is your negative two
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    okay so puerto ri nothing arranged guys
  • 00:03:08
    human numbers are adding set from lowest
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    to highest or highest lowest depending
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    guys
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    um
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    a relation in which each member of the
  • 00:03:24
    domain is paired to exactly one member
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    of the range is called a function
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    so on the banks of being none
  • 00:03:34
    [Music]
  • 00:03:35
    relation
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    function
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    if no two ordered pairs have the same x
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    value but different
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    y values
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    function
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    number one
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    we have one two two three three four and
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    four comma five
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    so as you can see guys um
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    input
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    nothing is a function
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    okay
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    next number two
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    we have one comma one
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    then two comma two three comma three and
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    four comma four
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    so as you can see guys now you mean
  • 00:04:45
    nothing is
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    a unique output
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    or is output so ebx bn young number to
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    nothing is also a function
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    okay
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    next number three
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    one zero
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    zero one
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    negative one zero and zero
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    negative one
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    okay
  • 00:05:12
    so guys
  • 00:05:20
    which is zero
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    and zero
  • 00:05:47
    domain is paired to exactly
  • 00:05:49
    one member of the range so this time
  • 00:05:54
    domain
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    is
  • 00:06:01
    okay which is one and negative one so
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    therefore
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    uh number three is not a function
  • 00:06:10
    okay
  • 00:06:12
    so next number four we have negative two
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    four
  • 00:06:15
    negative one one
  • 00:06:18
    zero zero one one then two
  • 00:06:22
    four
  • 00:06:23
    okay
  • 00:06:24
    so
  • 00:06:27
    uh
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    domain nothing which is negative two
  • 00:06:30
    negative one zero one two is
  • 00:06:43
    is a function
  • 00:06:48
    guys
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    okay next
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    uh functions can also be represented
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    through mapping
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    okay so where the elements of the domain
  • 00:07:00
    are map
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    to the elements of the range using
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    arrows okay so in this case
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    uh the relation or function is
  • 00:07:08
    represented by the set of all the
  • 00:07:10
    connections
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    by the arrows all right so try
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    which of the following mapping
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    diagrams represent function
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    x component
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    corresponds to a unique
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    range tama young one corresponds to
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    three two corresponds to five three
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    corresponds to nine then four to seven
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    then five to thirty three so latina
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    input nathan is my unique output so
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    therefore your number one nothing is a
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    function
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    okay so function n
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    so next number two
  • 00:08:08
    uh we have
  • 00:08:10
    x
  • 00:08:51
    u7 output near one you eat an output in
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    a zero then your nine and output is zero
  • 00:08:58
    all right so one problem guys
  • 00:09:02
    okay
  • 00:09:04
    so next number three naman
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    meru
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    11 13 17 19 and 23.
  • 00:09:18
    so guys
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    11 and
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    13.
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    okay then at the same time your input
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    not in the two is made in the output
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    output
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    okay so this time
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    uh your input not in the seven meet the
  • 00:09:59
    level output so eb sub n
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    uh this function or this relation is not
  • 00:10:04
    a function
  • 00:10:09
    all right nine indian but guys you're
  • 00:10:12
    adding uh
  • 00:10:13
    mapping diagrams
  • 00:10:16
    okay so i unmoved dials of functions as
  • 00:10:20
    a graph
  • 00:10:21
    in the cartesian plane
  • 00:10:23
    all right so given the graph of a
  • 00:10:25
    relation we can easily identify if it is
  • 00:10:28
    a function or not by using the vertical
  • 00:10:30
    line test
  • 00:10:32
    okay so underneath vertical line test
  • 00:10:35
    a graph of a mathematical relation is
  • 00:10:37
    said to be a function
  • 00:10:39
    if any vertical line
  • 00:10:42
    drawn passing through the graph
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    touches the graph at exactly
  • 00:10:46
    one point
  • 00:10:48
    all right so if it's a bn
  • 00:10:50
    uh magicking function is a graph if
  • 00:10:54
    i connect in a vertical line
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    that is
  • 00:11:12
    example so which of the graphs
  • 00:11:15
    represent a function
  • 00:11:18
    so letter a
  • 00:11:20
    so little guys are to test the graph
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    again the gamma line of vertical line
  • 00:11:26
    okay so um
  • 00:11:28
    in a vertical line so any point in graph
  • 00:11:34
    in your guys
  • 00:11:55
    represents a function
  • 00:12:00
    okay
  • 00:12:02
    so next number two
  • 00:12:04
    or letter b
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    so determination is straight line
  • 00:12:08
    so
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    this line represents a function
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    this ellipse is not a function or this
  • 00:12:48
    graph is not a function
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    so that means
  • 00:13:13
    uh this graph represents a function
  • 00:13:32
    um
  • 00:13:43
    so that means this type of hyperbola is
  • 00:13:46
    not a function
  • 00:13:53
    and so gangnam language simply guys give
  • 00:13:55
    me the new outing vertical line this
  • 00:14:01
    so this is the end of our video i hope
  • 00:14:04
    uh 19 day and you guys go on about ebay
  • 00:14:06
    subscribe
  • 00:14:09
    and if you have questions or
  • 00:14:11
    clarifications kindly put them in the
  • 00:14:13
    comment section below
  • 00:14:15
    thank you guys for watching this is prof
  • 00:14:18
    d i'll catch you on the flip side bye
Tags
  • Mathematics
  • Relations
  • Functions
  • Domain
  • Range
  • Vertical Line Test
  • Ordered Pairs
  • Mapping Diagrams
  • Function Graphs
  • Educational