Functions 9 - one-to-one Functions

>> In this video we'll be discussing one to one functions,
the definition determines if functions are one to one,
visual representations and how
to use the horizontal line tests.
This is part 9 of functions, and we're going
to be discussing the definition of a one-to-one function
and then determined if certain functions are one-to-one or not.
So here's the definition.
A one-to-one functions,
sometimes thee abbreviation is just 1-1, is a function
such that every input, which is usually the X value
if we're talking about ordered pairs corresponds
to only one output.
That would be the Y value of the ordered.
And every output, the Y value,
corresponds to only one input, the X value.
Okay, so I'm going to talk about broad terms
of one-to-one functions to begin with.
One example is the marriage function.
So you think of a society where polygamy is not allowed.
You'd say marriage is a one to one function.
Each wife has only one husband
and each husband has only one life.
So for instance, let's say you're at a party
and there are three couples there.
and if I wanted to write this as an ordered pair I can say well,
I've got let's say the wives' names are Amy, I'll call that A
for Amy and Beth, B for Beth, and C for Cyndi.
And let's say that Amy is married to Don,
put a D. Beth is married to Ed, so I'll put an E,
and Cyndi is married to Frank, which would be an F.
For instance, three couples that are married where the first,
the X coordinate is the wives, the letter of the wive's name
and the Y value set in coordinate is the first letter
of the name of the husband's name.
This is a one-to-one function because if you look
at all the ordered pairs there's no repeats in the X values.
C, A, B, C. There's no reaps
and there's also no repeats in the Y values.
So this would be a one-to-one function.
Now if you had Amy married
to two plus it's not going to work, right?
So if we had Amy also married to someone named Gary, okay,
that's a polygamist type of society,
this would not be one to one.
In fact this would not even be a function because remember
in a function the X coordinates can't be listed more than once.
For each X there's only one Y,
but it doesn't have to go the other way.
So this is not one-to-one, and it's not even a function at all.
Now what about if you had this situation,
there's a polygamist situation where Amy
and Beth were both married to don.
This is not one-to-one but is a function.
All right, so remember
in a function the X values can't be repeated.
For X value there can only be one Y but it doesn't have
to go the other way around.
Let's use some ordered pairs now.
All right, here's an example of three functions,
and let's decide if they're one to one or not.
All right, let's see number one.
Really you have to look at the Year values
and they can't be repeated
because if it's a sanction we already know the X values cannot
be repeated so notice for number one for the X values are 3, 0,
-1, no repeats, but for the Y values I've got - 1,
4, 5 and 3, no repeats.
So yes, this is one to one.
Let's look at the next one.
I already know it's a function
because I've said these are functions.
So I'm basically going to look at the Y values.
Now the Y values I've got three, I've got four, I've got three.
There's a problem here that there is a repeat.
So I have two different X values
that have three four those output, so no.
This is not one to one.
All right, this is I'm going to call the mother function.
So you've got the name of the child
and the name of the mother.
You want to know this assumption.
So Tom's mom is Lisa, C's mom is Mary, Jen's mom is Lisa
and Ed's mom is Cindy.
Is this a one-to-one function?
Nope, because these two, Tom and Jen, Lisa for their mother
so it's not one to one.
So there's an example.
You could also do these problems by looking
at a little graph this way.
These are the same three functions
where I just used a visual to show where the input is sent
to for the output and look at number one.
There's four elements in each, and each geese
to exactly one element and this was the one to one function.
Now here's a problem with number two and three.
Notice that for the input there's three elements
but there's only two in the output for number two,
and for number three there's four in the input
and three in the output.
And so necessarily what's going to happen is
over on the right side you're going to see an arrow going
to -- more than one arrow going to the same number,
and that's what's happen.
And that's why each one of these are not one to one.
And that's the reason, so it's a visual representation.
In order for it to be one-to-one you have to have the same number
of elements in the input but the output
so that's one way of telling.
And then you also have to make sure none are repeated.
Another way to tell if a function is one to one or not is
by looking in its graph.
Now when we talked about whether something was a functions we
used the vertical line test.
So the vertical line test tells us if something is a sanction
and we remember what that meant, if I had something like this,
I have to make sure that no matter I put a vertical line I
can't cross the function in more than one place
because that would mean there were two different Y values
for one X value.
So this graph I've put up here.
I'll call it F, does pass the vertical line test.
We're going to erase theses now.
Now I want to see if it's one to one.
Now F is a section but the question is it one to one.
You did the horizontal line test
so it means wherever you draw a horizontal line you can't go
through the function more than once.
And here's a problem.
I go through it in two places.
That means like there's some ordered pair here
where the X value is the same.
So whatever the X value is I don't know.
I'm sorry, I meant to say where the Y value is the same, right?
It's up at the same height, so the Y value, whatever it is,
is exactly the same for whatever these two X values are.
These are two different X values.
Also if this part right here is completely horizontal,
if you have a horizontal line in your function,
definitely it's not going to pass it.
So that's how to use the horizontal line test.
So let's look at another example.
Let's say we had something like this,
so this is the graph of some function.
First of all, is it a function?
And it is, it will pass the vertical line test,
but will it pass the horizontal line test.
And you can see now that there's a problem
so this is not one-to-one.
Let's try one more.
Let's take this graph, let's call it H,
determine if this is a function and if so if it's one to one.
Well, first let's look at the vertical line test.
Yes, no matter where you put a vertical line it's not going
to go through it more than once, so yes this is a function,
definitely, and does it also pass the horizontal line test?
Yep, no matter where I put it passes
so this is a one-to-one function.
All right, the next figure we're going
to be talking bout inverse functions, okay?
And you could only be talking about inverse function
if it's one to one so that's exactly why you have
to know what a one to one function is.
?? ?? ?? ??