Welcome to MySecretMathTutor. For this video we are going to talk about
how you can determine if a function is one-to-one. Now of course if we are going to determine
one-to-one for functions we first have to really look at what that means.
We say that a function is one-to-one if every input goes to a unique output.
And the way you want to look at this definition, is it's like a little bit of extra criteria
that we put on functions. So for a relation to be a function you want
to make sure every input goes to one output, and this is just making sure that that particular
output is unique, nothing else goes to it. So to make this idea a little bit more concrete,
let's look at an example of a function versus a one-to-one function.
Here in a function we have particular inputs and they are mapped to particular outputs,
and this is a function, because every time I have a particular input it goes to one output.
Now you may notice that down here, both the -3 and the 2 go to 4.
Now this is not a problem with being a function but this type of situation makes it not a
one-to-one function, because we have two inputs that go to the same output.
Now let's go ahead and turn our attention to this guy over here, and again you'll notice
that we have different inputs mapped to different outputs, but you know what, every single input
only goes to one unique output. That's what makes something a one-to-one function.
Now in practice how to you really determine if something is one-to-one, or not?
Well, if you have a graph to look at, then you can determine if its one-to-one by using the
vertical and horizontal line test. Each of these test do something different,
but let me see if I can break them down. If you use the vertical line test, on say
a relation, this is testing if it is a function. If every input goes to one particular output.
When you use the horizontal line test, now you're checking those outputs to see if there
are two inputs that go to the same spot. And the key with using either of these test,
is that you either imagine a vertical or horizontal line, and you want to check to make sure
that it does not cross in more than one spot. So let's use this test on a couple of generic
functions and see if they are one-to-one. So here is an example. I have this function,
or at least I have this relationship and I want to test if it is a function.
So you want to imagine a vertical line, and what you are testing is: no matter where you
put this vertical line, does it cross in more than one spot.
Well you can see that, you know what, no matter where I put this vertical line, it only crosses once.
So we can say, yes, this is a function.
Now to take a little bit of an extra step, and say, well is it a one-to-one function?
Now we are going to use the horizontal line test, and say, OK, if I put a horizontal line
on here, does it ever cross in more than one spot?
And it looks like it does. So if it does cross in more than one spot,
then you say, no. It is not a one-to-one function. Alright, let's try this again with another
generic function. Something like this.
So first we could test it and say, is this relation a function?
Let's use our vertical line test. We're checking to see if it crosses in more than one spot.
Doesn't look like it does. So we will say, yes.
Now let's see if it is one-to-one. So now we are using the horizontal line test.
I'm seeing if it ever crosses in more than one spot.
Looks like that is not the case. So we can say, yes, this is a one-to-one function.
Now its really nice to have a graph and apply both of those test, but sometimes you may
just be looking at an equation and you still have to kind of determine is this thing one-to-one
or not? If you don't have the graph, you can determine
if a function is one-to-one by checking a particular output and seeing if it can be
obtained from say two different inputs. Now what this is really a test for is not
if it is one-to-one for sure, but really a test that it is not one-to-one.
So your are kind of looking for a counter example that breaks it in some sort of way.
So for example, maybe I'm looking at this guy.
f(x) = (x - 2)^2 + 3 And it does happen to be a function.
But you know, I'm curious, is it a one-to-one function?
Well here is one way I can show that it is not a one-to-one function.
I can test two particular inputs. For example, let's go ahead and test it out
a 1, and let's see the result we get. So (1 - 2)^2 + 3
That would equal a (-1)^2 + 3 Or 1 + 3 = 4
So when I put in a 1, the output is 4. Now let's try for the 3.
We're going to put that in there as well. (3 - 2) is a positive 1
So it looks like we get the same output. So this is the problem we are looking to avoid
with one-to-one functions. I have one particular output, but two different
inputs go to the same spot. So we say that this is not one-to-one.
So, you know that doesn't necessarily help as much with finding one-to-one functions, but it
is a great way to test if something is not one-to-one.
So remember the key here with determining if something is one-to-one is really by looking at inputs
and outputs and figuring out if every input goes to a unique output.
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