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One to one functions

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Hi, and welcome. My name is Anthony Varela, and today we're going to talk about one to one functions. So we'll talk about the characteristics of one to one functions. We'll look at a graphical test to determine if a function is one to one, and an algebraic test to determine if a function is one to one. So let's first start off by talking about functions and how we use what's called the vertical line test to determine if a graph of a relation is a function.

And so with our vertical line test, we have a graph of what we think might be a function, and we put this vertical line, we scan it through the entire graph, and we're looking to make sure that the graph only intersects the vertical line at one point across the entire graph. And if it does, that means it's a function.

So this graph here is the graph of a function. And what this means then is that there's only one y value and exactly only one y value for any given value of x. So that's how we determine if a relation is a function. But what about one to one functions?

So first, one to one functions have to pass the vertical line test because it has to be a function. But what makes the function one to one? While the vertical test verifies that there is exactly one y value for any given value of x, one to one functions the opposite is also true. So there is exactly one x value for every given value of y. So the relationship between x and y is one to one.

So we can use a horizontal line test to determine if a function is one to one. And the horizontal line test works in a very similar way as the vertical line test does. We just pass a horizontal line across the entire graph, and we look to see and make sure that the function only intersects this horizontal line at one point across the entire graph. And here we see that it does. So maybe it might be easier to see what a one to one function doesn't look like.

So first, we can pass this through the vertical line test to confirm that this is indeed a function. So I don't see any point were the graph intersects the vertical line more than once. So it's a function. But is it one to one? So now we'll pass this through the horizontal line test.

And so far, so good until we get to here. We see that the graph touches the horizontal line more than once here and here. And it happens a couple of different times. It actually hits the graph one, two, three times here. So this is not a one to one function. It's a function, but it's not one to one.

So let's determine if this function f of x equals 5x minus 3 is one to one. Now, algebraically what we can do is assume that f of a equals f of b. And then what we're going to do is prove that a equals b.

So what I mean by assuming that f of a equals f of b, we're going to write the definition of our function using a and b. So f of a would be 5a minus 3, and f of b would be 5b minus 3. We're just replacing x with a and with b and setting these two equal to each other.

Now what we have to do is prove that a equals b. And more specifically, a always equals b and only b. And so we can see we're just going to do some inverse operations here and try to isolate a and isolate b. So I can add 3 to both sides of this equation and I get 5a equals 5b. Then I will divide the equation by 5, and then I get a equals b. So I've proven that a equals b. So f of x equals 5x minus 3 is an example of a one to one function.

Let's take a look at another example. We want to determine if g of x equals x squared minus 4 is one to one. Now we're going to do this using our algebraic technique as well. So we're going to assume that f of a equals f of b. So what that would mean then is that a squared minus 4 equals b squared minus 4. And now we're going to prove that a equals b, and if we can do that, it's a one to one function. If we can't, then it is not a one to one function.

So I'm going to add 4 to both sides of this equation. I get a squared equals b squared. Well, to undo the square, I'm going to take the square root. So taking the square root of both sides, I get plus or minus a equals plus or minus b, because in equations when we take the square root, we always include plus or minus.

But this is tricky because I'm tempted to say, great, this means that a equals b. But there are a couple of possibilities here with that plus or minus. So it certainly means that a could equal b. It means that negative a could equal negative b. But there are two other possibilities with this plus or minus on both sides of my equation.

This also means that negative a could equal positive b, and negative b could equal positive a. And that's not proving that a always equals b, that there's a one to one relationship between a and b. So this is actually not a one to one function. It's a function, but not one to one.

So let's review this lesson on one to one functions. Well, every element in the domain corresponds to exactly one element in the range. That's what defines a function. And with one to one functions, the opposite is also true. So every element in the range corresponds to exactly one element in the domain. That's a one to one function.

So we can use a vertical line test to determine if a relation is a function, and we can perform the horizontal line test to determine if a function is one to one. So one to one functions have to pass both the vertical and the horizontal line test. Algebraically, you could assume that f of a equals f of b, and if you can prove that a equals b in all cases, then you have a one to one function.

So thanks for watching this tutorial on one to one functions. Hope to see you next time.