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One to One Functions

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Today, we're going to talk about 1 to 1 functions. So we're going to start by reviewing what it means to be a function, and then we'll talk about what it means to be a 1 to 1 function, and we'll do some examples. So let's start by reviewing what it means to be a function.

I've got an example of a function, f of x is equal to x squared minus 4x plus 7, and its corresponding graph. So remember that the definition of a function is that every element or input in the domain corresponds to exactly one element in the range or output. So by looking at the graph, we can tell whether a relation is a function or not using something called the vertical line test, which says that if we were to draw a vertical line anywhere through our graph, it should cross the graph at exactly one point, more than one point.

So we can see that this graph is a function because no matter where we draw a vertical line, it's only going across our graph one time. And we know that passing the vertical line test means that there's exactly one y value. So for example, at this point there is exactly one y value for this given x value. So we know, again, that by passing the vertical line test there's only one y value for any given x value.

So now, let's talk about functions that are 1 to 1. Functions that are 1 to 1 will pass not only the vertical line test, but also the horizontal line test. So here's an example of a function that is 1 to 1.

f of x is equal to x minus 2 and its corresponding graph. So we can see that this function passes the vertical line test, first of all, because no matter where I would draw a vertical line, I can see that it's only ever going to cross my graph one time. So this passes the vertical line test. But I can also see that it passes the horizontal line test because no matter where I would draw a horizontal line through my graph, it's only ever going to cross my graph one time at any given value for y.

So this function also passes the horizontal line test. And again, because it passes the vertical line test and the horizontal line test, we call that a 1 to 1 function. So in 1 to 1 functions, not only does every value or element in the domain correspond to exactly one element in the range, but every element in the range corresponds to exactly one element in the domain.

So let's do an example to determine if a function is 1 to 1 So I have this function, f of x, which is equal to negative x squared minus 4x minus 1 and its corresponding graph. I know that if this function is 1 to 1, it needs to pass not only the vertical line test, which tells us that it's a function, but also the horizontal line test, which would tell us that it's a 1 to 1 function.

So I already know because I know it's a function that it's going to pass the vertical line test. But I can see that that's true because no matter where I draw my vertical line, it's only going to cross the graph one time. So to see that it's a 1 to 1 function, I want to see that it passes the horizontal line test.

So if I were to draw a horizontal line anywhere through my graph, it should only cross one time. But for example, when I draw my horizontal line here, I see that it crosses my graph twice, which means that this is not a 1 to 1 function because it does not pass the horizontal line test.

So let's do some examples determining whether a function is 1 to 1 algebraically. So I have the function, f of x equals 7 x minus 5, and I want to show that this is a 1 to 1 function. So I start by assuming that f of some value a is equal to f of some value b. And if this is a 1 to 1 function, then that means that a is equal to b.

So I can use this assumption and write f of a by substituting my x variable with a. So this will become 7 times a minus 5. And f of b will be 7 times b minus 5. I can simplify this by adding 5 on both sides. And I see that it's going to cancel out the minus 5 on both sides so I'm left with 7a equals 7b.

Then I can divide by 7 on both sides. And again, that's going to cancel on both sides and I'm left with a is equal to b.

So starting with my assumption that f of a is equal to f of b because this f of x is a 1 to 1 function, I can also show that a is equal to b. So let's do one more example to determine if a function is 1 to 1.

I've got the function, g of x, which is equal to x squared minus 3. So again, I'm going to start by assuming that g of a is equal to g of b. And I want to show that a is equal to b. So g of a is going to be equal to a squared minus 3.

And g of b is going to be equal to b squared minus 3. So I can simplify this by adding 3 on both sides, which will cancel out the minus 3 on both sides, which would give me a squared is equal to b squared. So it might seem that if a squared equals b squared then that means that a is equal to b, but we actually have four different possibilities here.

If a squared is equal to b squared, that means that a can be equal to be, positive a equal to positive b. We can have negative a be equal to negative b. In which case, a squared would still be equal to b squared.

We can have positive a be equal to negative b, which, again, here a squared would still be equal to b squared. Or we could have negative a is equal to positive b, which, again, would lead us to a squared be equal to b squared. So from a squared equal to b squared, we can end up with one of these four possibilities, which means that a is not always equal to b. We could also have one of these three scenarios. And so we cannot always say that a is going to be equal to b for this function, g of x. So we're concluding that because a is not always equal to b that this function, g of x, is not 1 to 1.

So let's go over our key points from today. 1 to 1 functions not only pass the vertical line test, but they also must pass the horizontal line test. Passing the vertical line test means that there's exactly one value for y at any given x value and passing the horizontal line test means that there's exactly one value for x at any given y value. A function is 1 to 1 if assuming f of a equals f of b, a equals b.

So I hope that these key points and examples helped you understand a little bit more about 1 to 1 functions. Keep using your notes and keep on practicing and soon you'll be a pro. Thanks for watching.