Welcome back, everyone. In the last video, we talked about relations and functions, and we specifically focused on graphs. Now, recall in the previous video that we had this special tool called the vertical line test where we could see if a vertical line ever passed through more than one point anywhere we would draw it on the graph. If this ever happened, we would say that we failed the vertical line test, in which case we would not be dealing with a function. However, if this did not happen, then we were dealing with a function. We talked about all of this in the last video. Now in this video, we're going to be taking a look at how we can verify equations as functions and there is a test for this, but the steps are a little more complicated. So pay close attention because I am going to walk you through them.
The first thing you want to do when dealing with an equation is always solve for y. This should be your first step when you have an equation. So if you have an equation and you've solved for y, what you then want to do is see if any x's will result in multiple y's. If this does happen, then you are not dealing with a function. However, if this does not happen, then you are dealing with a function. Now, notice for both of these tests, whenever we get an answer of yes, that means it's not a function. But whenever we get an answer of no, it is a function. So that's just something you can use to remember how these tests work. Now, let's actually put this test to use and solve an example.
So here we have these two equations, and we want to see whether or not either of these equations are functions. We'll start with equation a, which is this one on the left. We have y + 4 = 3x and our first step should always be to solve for y. So I'll take this 4 and subtract it on both sides of the equation to cancel the fours on the left. That will give me that y is equal to 3x - 4. And now that we've done this, we have solved for y. So the next thing I'm going to do is I'm going to try a bunch of x values to see if we have a situation where any of these x's result in multiple y's. So let's try an x value of 0. So we'll have that y is equal to 3 × 0 - 4. 3 × 0 is 0, and 0 - 4 is -4. Now let's try an x value of 1. We'll have y is equal to 3 × 1 - 4, 3 × 1 is 3, and 3 - 4 is -1. Now let's try a negative number like -1. So, we'll have y is equal to 3 × -1 - 4, 3 × -1 is -3, and -3 - 4 is -7. Now let's try an x value of 2. We'll just try one more value here to see what happens. So, we have 3 × 2 - 4, 3 × 2 is 6, and 6 - 4 is 2. Now notice what happens here. For every x value that we replace this x with, we always get only 1 y value as an output. And because of this, we could say that this is an example of a function because we only get 1 y value for each x value. And you may notice that this is actually a familiar form that we have this equation. We have the form y = mx + b. This is the slope-intercept form of a line that we've talked about in previous videos. So whenever you are dealing with a line that has a defined slope, and in this case, the slope would be positive 3, then you're always going to be dealing with a function in this situation. So that's just something you can remember when dealing with these types of problems. But now let's take a look at this equation on the right side. So for equation b, we have x² + y² = 25. And just like before, I'm going to solve for y. So I'll subtract x² on both sides of the equation and that'll get the x²s to cancel on the left, giving me that y² is equal to 25 - x². Now my next step here is going to be to take the square root on both sides of the equation, giving me that y is equal to ±√(25 - x²). So we've now solved for y, so this is the equation that we have. Now our next step is going to be to test out a bunch of different x values. So just like before, I'll start with an x value of 0. So we'll get that y is equal to ±√(25 - 0²). 0² is just 0, so all we're going to end up with is ±√25. And recall the square root of 25 is just 5. So because of this ± sign, we're going to end up with positive 5 and negative 5. But this is interesting. Notice how one x value gave us multiple y values. And we said whenever this happens that we are not dealing with a function. So, in this example, we can see that this is not a function because we already have an x value that gives us multiple y values. And something else you can remember with this equation is notice that we started with y². Whenever you have an equation where y has an even power that it's raised to, then it's always going to be not a function. So that's just something to keep in mind. Now neither one of these situations are scenarios that you're going to have to memorize, but it's just something that can help you if you want a quick shortcut to figuring out whether an equation is a function.
Now there's one more thing I want to mention before finishing this video and that is if you have an equation that is identified as a function, you can rewrite it using function notation. And when you use function notation, what you want to do is replace the y with f(x). So, for example, in this scenario that we had before where we had y = 3x - 4, you could take this y at the start and you could replace it with f(x) where x represents the inputs and f(x) represents the outputs. You could not, however, use this function notation in the other example because this was not identified as a function. So this is how you can verify equations as functions. Let me know if you have any questions and let's move on.