Uploaded by videosbyjulieharland on 10.07.2010

Transcript:

>> 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.

?? ?? ?? ??

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.

?? ?? ?? ??