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Relations & One On One, Onto, Discrete and Continuous functions
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Feb 19, 2025
In Algebra 1, we were quickly introduced to relations and functions and learned how to identify the domain and range for several types of functions. Now we'll learn how to analyze relations and functions , we'll also learn how to identify one on one, onto, discrete and continuous functions and briefly talk about the vertical and horizontal line tests. Chapters: 00:00 Introduction 00:35 Example of a function 02:46 One One One Functions 04:23 Onto Functions 05:24 Onto and One on One Functions 06:35 Discrete functions 07:09 Continous functions 08:05 Using The vertical line test to identify functions 11:25 Working on more examples to close the section
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0:00
how that so we're going to start um
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relations and functions so if you did
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algebra one and pre- algebra you were
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taught about functions right now a
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function by
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definition is a relation in which each
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element in the
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domain is paired or linked with exactly
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one element in the range so a function
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has a domain and a range and the domain
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are all the X values and the range are
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all the Y values right so function so
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example of a function marriage for
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example that's a function right cuz each
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man is linked to exactly
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what one what
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woman right so that is a function but
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you don't have one man two women within
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the Christian marriage that's not a
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function in the Christian marriage is
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one man two one woman right we don't
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have one to two so that's a function so
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this is a function
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okay here in as an example one is linked
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to what five and this is the set of the
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domain this is the domain these are all
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the X values right and range the Y
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values they also called The Domain is
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also called input i n p u t right
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input and the range is called out put
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right the range pretty much depends on
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the input right what you put in what you
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get depends on what you put in right so
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one is linked to five two is linked to
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six and then three is linked to eight
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that is a
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function okay this is a function now if
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you go here for example this is not a
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function that is not a function because
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one right is in the domain it's linked
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to what six and one is also linked to
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what seven seven so that is not a
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function right because for this to be a
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function it has to be that one is link
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to six two is link to eight and then
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three is link to nine in this case that
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would be a function but if you do this
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then that can't happen that's not a
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function and like I say if you have a
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man with two wives now although in
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biblical times we had those things that
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was not original God's original plan he
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a Jesus talks about that in the Book of
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Matthew right so it's one man one wife
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one here is linked to six and 7 not a
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function does that make sense right
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those are called now this is just a
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function now we're going to talk about
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one onone function a one onone function
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now in a oneon-one function as you can
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see here each element in the domain or
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all the X values is linked to exactly
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one element in the the range all right
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that's a one1 function all right so here
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we have three
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10 4 11 right sorry Mr can I interrupt
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you for a second back one1
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fun in a one1 function right each
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element in the domain
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is linked to exactly one element in the
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range right here you have 310 411 712 it
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does not matter that eight is by itself
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right we can call eight an unmarried
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woman that's
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fine she's sitting on island here on
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Island right that's fine right as long
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as as long as the man as long as the man
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was in the picture JD was not the
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picture free to as long as the men are
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linked to each one wife that's okay we
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can have a woman on the island here we
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are fine that's what it means so this
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not a oneone function but the poor
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woman's all po woman that's
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problem
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right we we get to have an upper hand on
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that all right so now this is a one1
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function now here we have what you call
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it on two function right on two function
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in this case you can see that it's okay
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now we're not going to use this as an
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example because in this on two function
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it violates our very standard of
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Christian marriage because here 34 right
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is linked to 10 that doesn't work that's
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two people married one person that's not
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working right the onto function does not
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apply so we not going to use that
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example but I still in this case each
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like each element in
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the uh in the range is linked to an
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element in the domain you see that right
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for example you have 10 is linked to
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three and four 11 is linked to six and
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then 12 is linked to eight every single
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one of those element in the range must
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have something must have an element or
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two or more in the in the domain does
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that make sense for the function to be
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onto right so again I'm not going to use
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this as an example because that's not
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that doesn't qualify now for a function
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to be on two and one one at the same
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time right you have to
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have a single
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relationship here right three is linked
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to what 10 four is linked to 11 six is
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linked to 12 8 is link to 13 this is
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exactly the best representation of
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Christian marriage because you have each
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man to his wife each man to his wife
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each man to his wife wait so the one
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above it can't it doesn't work work
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because you see here right we have B
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which is an in the range is link to
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three and four that's not that's not a
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one one function is a onto function it's
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still a function but it's an onto
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function right but within the sacredness
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of Christian marriage this is what we
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want we want one one we want one one and
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on two at the same time where these two
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meet so we have three is linked to 10
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four is linked to 11 six is linked to 12
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8 is linked to 13 so this in this case
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is called on Two and a one one function
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does that make sense right so these are
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the cases now we also going to talk
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about discret function funs and
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continuous functions by definition this
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is really super easy a district function
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is a relation in which the domain is a
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set of individual points right a scatter
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PL for example will be called a discrete
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function because there's no continuity
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you see that right H one is just
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scattered it's a DOT a DOT a DOT and a
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DOT this is called a discrete function
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that makes sense discret d i
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s e disc function do I have to do the CL
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say it again do I have to do the CL yeah
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right and then and then you have also
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where we have a continuous function
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right in this case in this case the
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domain is an infinite number of point
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right if you look here the domain is on
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your X AIS so the domain in this case is
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an infinite number of point because if
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you trying to count how many points you
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have on this curve and this line you're
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not going to stop you I'm going to be
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here after 45 here you still be trying
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to figure out how many points you have
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here because it's infinite so there's
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continuity here this is called a
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continuous function so there's Infinity
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there's Infinity an Infinity number of
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uh points on the domain meaning if I was
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to like go here right I'm not going to
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stop because the point indicates that
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the function keep going right what would
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be your domain you don't know cuz you're
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going to keep going all the way to
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Infinity so there's no stopping because
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there's an infinite number of point so
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therefore this is called a continuous
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function
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right now we also going to learn about
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the vertical line test and the vertical
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line test is can I eras this yeah all
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right the vertical line test I'm going
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to show you how that
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works the vertical line test it is used
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to figure out what a function is one on
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one or actually no the vertical line is
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to figure out if a relation is actually
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a function okay it's called a vertical
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line vertical line test right my
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mar vertical
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line test right so basically I have a
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curve right if you look at this curve
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here right what kind of curve is this I
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don't know if you guys learn this in
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algebra one what do we call this yeah
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but I know how to do the vertical right
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what kind of yeah what kind of curve is
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this no you it's a paraba right right a
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parabola this is called
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vertex a parabola p a
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parabola right a problem right it's not
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a u ter right so now how do you know if
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this is this is a function you draw a
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vertical line right if I draw a vertical
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line across this curve how many time am
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I crossing the curve one because I'm
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crossing only one time this qualifies as
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a
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function right so therefore this a
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function it means that it passes the
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vertical line test it passed it right if
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I draw another line here it's only one
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time here if you draw an infinite number
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of vertical line through this is only
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going to cross this curve one time so
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this therefore this is called a function
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right wait what if you go this way now
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good question if you go this way how
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many I cross it poly relationship and it
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does not work no this is how many times
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I cross it twice right because I cross
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this twice this is not a one1 function
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it's not a one1 because I cross this
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twice this is called a horizontal line
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test horizontal line right if you draw a
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horizontal
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line right and you cross this twice it
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doesn't qualify so therefore this is no
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longer a one1 function it is a function
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but is not one1 right does that make
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sense yes
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question you just draw imaginary curve
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with your hand yes I am all right so
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what about this function here is this
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101 is this a function actually is this
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a function
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noce is it so if it's not a function we
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don't even need to figure out if this is
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actually one1 because it's not even a
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function so therefore it doesn't even
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matter right so now we're going to work
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on some uh problems in the book and
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we're going to try and figure this out
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so let me give you an example and we
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going to try and see if this is one one
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onto or
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neither so this is our relationship here
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right so let's actually let's create a
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relationship here and we going to try to
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figure out this relationship is one1 or
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I
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why am I
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there can't do
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this I don't want to
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be oh my
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God horrible I'm not
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let try to see there I'm
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[Music]
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single anybody
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else hold on hold on on what's going on
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here
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is this is he one on one no is he on to
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no no
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nothing you
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forgot Mary and Mary
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also and
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by you forgot Nicholas and
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jich is that a husband in-law
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again I don't want
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to why am I
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always ginger on there
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[Laughter]
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[Applause]
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101
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[Music]
15:10
[Music]
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seconds
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[Music]
15:40
[Music]
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wait a minute so let's let's build this
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here right so first we need to identify
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the domain D is the domain and R is a
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range right
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range and the domain so the question is
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asking to figure out whether each
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mission is a function number one and
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then if it is a function determine if it
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is one one onto both or neither right so
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first we need to identify the domain the
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domain is -6 -5
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-31 and 6 and then the range would be -1
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-9 -7 7 and9 right9 is twice we don't
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need to add it twice we only put it once
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okay so now we're going to build it
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using our little diagram here right so
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the domain and the range so it's
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-65 -3 -1 and then
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six9 right so6 is link
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to15 is link
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to9 -3 is link to this one1 is linked to
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7 and then six is L
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to9 now what kind of function do we have
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here is it a function number one yes or
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no
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no it's a function it's a
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[Laughter]
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function so we just going to finish this
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one real quick all right so for this
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specific case the function was what on
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two right because -9 was linked to
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what5 and then six right so this is are
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onto function here this is what we are
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he a function but it's two it's not one
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on one because you don't have one here
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and then one here we don't have a one
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one because if you look here
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right9 is linked to
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what5 and six so this is on two function
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okay so all right so this is going to be
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uh that's the example so what I want to
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do now is I want to
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