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# Relations and functions

Video transcript

Is the relation given by the
set of ordered pairs shown below a function? So before we even attempt
to do this problem, right here, let's just remind
ourselves what a relation is and what type of relations
can be functions. So in a relation, you
have a set of numbers that you can kind of view as
the input into the relation. We call that the domain. You can view them as
the set of numbers over which that
relation is defined. And then you have
a set of numbers that you can view as the
output of the relation, or what the numbers that can
be associated with anything in domain, and we
call that the range. And it's a fairly
straightforward idea. So for example, let's say that
the number 1 is in the domain, and that we associate the
number 1 with the number 2 in the range. So in this type of
notation, you would say that the relation
has 1 comma 2 in its set of ordered pairs. These are two ways of
saying the same thing. Now the relation can also
say, hey, maybe if I have 2, maybe that is associated
with 2 as well. So 2 is also associated
with the number 2. And so notice, I'm just building
a bunch of associations. I've visually drawn
them over here. Here I'm just doing
them as ordered pairs. We could say that we
have the number 3. 3 is in our domain. Our relation is
defined for number 3, and 3 is associated with,
let's say, negative 7. So this is 3 and negative 7. Now this type of
relation right over here, where if you give me any
member of the domain, and I'm able to tell you exactly
which member of the range is associated with it, this is
also referred to as a function. And in a few seconds,
I'll show you a relation that
is not a function. Because over here, you pick
any member of the domain, and the function really
is just a relation. It's really just an
association, sometimes called a mapping between
members of the domain and particular
members of the range. So you give me any
member of the domain, I'll tell you exactly which
member of the range it maps to. You give me 1, I say, hey,
it definitely maps it to 2. You give me 2, it definitely
maps to 2 as well. You give me 3, it's definitely
associated with negative 7 as well. So this relation is both a--
it's obviously a relation-- but it is also a function. Now to show you a relation
that is not a function, imagine something like this. So once again, I'll
draw a domain over here, and I do this big, fuzzy
cloud-looking thing to show you that I'm not
showing you all of the things in the domain. I'm just picking
specific examples. And let's say that this big,
fuzzy cloud-looking thing is the range. And let's say in this
relation-- and I'll build it the same way that
we built it over here-- let's say in this relation,
1 is associated with 2. So let's build the
set of ordered pairs. So 1 is associated with 2. Let's say that 2
is associated with, let's say that 2 is
associated with negative 3. So you'd have 2,
negative 3 over there. And let's say on top of
that, we also associate, we also associate 1
with the number 4. So we also created
an association with 1 with the number 4. So we have the ordered
pair 1 comma 4. Now this is a relationship. We have, it's defined
for a certain-- if this was a
whole relationship, then the entire domain is
just the numbers 1, 2-- actually just the
numbers 1 and 2. It's definitely a relation, but
this is no longer a function. And the reason why it's
no longer a function is, if you tell me,
OK I'm giving you 1 in the domain, what member of
the range is 1 associated with? Over here, you say, well I don't
know, is 1 associated with 2, or is it associated with 4? It could be either one. So you don't have a
clear association. If I give you 1 here,
you're like, I don't know, do I hand you a 2 or 4? That's not what a function does. A function says, oh,
if you give me a 1, I know I'm giving you a 2. If you give me 2, I
know I'm giving you 2. Now with that out of
the way, let's actually try to tackle the
problem right over here. So let's think about its
domain, and let's think about its range. So the domain here,
the possible, you can view them as x
values or inputs, into this thing that could be
a function, that's definitely a relation, you could
have a negative 3. You could have a negative 2. You could have a 0. You could have a, well, we
already listed a negative 2, so that's right over there. Or you could have a positive 3. Those are the possible values
that this relation is defined for, that you could
input into this relation and figure out what it outputs. Now the range here, these
are the possible outputs or the numbers
that are associated with the numbers in the domain. The range includes 2, 4,
5, 2, 4, 5, 6, 6, and 8. 2, 4, 5, 6, and 8. I could have drawn this
with a big cloud like this, and I could have done this
with a cloud like this, but here we're showing
the exact numbers in the domain and the range. And now let's draw the
actual associations. So negative 3 is associated
with 2, or it's mapped to 2. So negative 3 maps
to 2 based on this ordered pair right over there. Then we have negative
2 is associated with 4. So negative 2 is
associated with 4 based on this ordered
pair right over there. Actually that
first ordered pair, let me-- that
first ordered pair, I don't want to
get you confused. It should just be this
ordered pair right over here. Negative 3 is associated with 2. Then we have negative 2-- we'll
do that in a different color-- we have negative 2
is associated with 4. Negative 2 is associated with 4. We have 0 is associated with 5. 0 is associated with 5. Or sometimes people
say, it's mapped to 5. We have negative
2 is mapped to 6. Now this is interesting. Negative 2 is already
mapped to something. Now this ordered pair is
saying it's also mapped to 6. And then finally--
I'll do this in a color that I haven't used yet,
although I've used almost all of them-- we have
3 is mapped to 8. 3 is mapped to 8. So the question here,
is this a function? And for it to be a function
for any member of the domain, you have to know what
it's going to map to. It can only map to one
member of the range. So negative 3, if you put
negative 3 as the input into the function, you know
it's going to output 2. If you put negative 2 into
the input of the function, all of a sudden
you get confused. Do I output 4, or do I output 6? So you don't know if you
output 4 or you output 6. And because there's
this confusion, this is not a function. You have a member
of the domain that maps to multiple
members of the range. So this right over here is not
a function, not a function.