Algebra

Functions as graphs | Functions and their graphs | Algebra II | Khan Academy

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Let’s solve some problems with points and feature graphs. In the task required of us to find the coordinates at each of these points. Let’s do it. Let’s start with this point here. Let’s call point a. We start by finding the x coordinate. x means how far to the left or right of the start point is. And it is 1, 2, 3, 4, 5, 6 to the left, or just -6 from the beginning of the coordinate system. Minus 6 is right here. And then the y-coordinate, which means how high the point is, right here. This is 4. Minus 6 and 4. Let’s do the same for point b. For the x-coordinate just go down, and that’s 7. And the y-coordinate, how high the point is, that’s 6. Okay. Let us now find the coordinates of point c. the x-coordinate is minus 8. 8 to the left of the beginning of the system. minus 8. And the y-coordinate is 2 squares below the beginning of the system. So the y-coordinate is minus 2. That is, nothing complicated. Point d. Point d, its y-coordinate is 4. No, sorry, its x-coordinate is 4, and then its y-coordinate, how much lower than the beginning of the system is, or just below the vertical axis, it is minus 7. And finally we are at point f. Choosing a nice color for the item is. Right there. Its x-coordinate is 5. On the abscissa axis x is equal to 5. And its y-coordinate, well, it’s neither above nor below the beginning of the system or the abscissa, so its y-coordinate is 0. We can draw a line and get right to 0. And so. We found all these coordinates. Now, in task 5 we need to find out if any dependency is a function. Here we have a trick that an addiction is not a function if there are two values \u200b\u200bfor x. For example. If I write that f (x) is equal to 5, if x is equal to 1, or is equal to 6 if x is equal to 1. This makes no sense. Why doesn’t it make any sense? Because if I put the number 1 in there, I don’t know what I will get a result. Will I get 5? Or will I get 6? This is a poorly constructed feature. This is not a feature. So if we have a similar argument situation, which gives us several different possible results, so this is not a function. Let’s see if there’s anything like that here. We are first given that if x is 1, then y is 7. If x is 2, y is 7, well. Yes. We can have two values \u200b\u200bfor x for one for y. For example, it would be the same to say that f (x) is equal to 7 if x is equal to 1 or 2. Everything is fine. For two different values \u200b\u200bof x we \u200b\u200bcan have the same result, but we cannot have two different values \u200b\u200bfor x to give us the same, excuse me, i.e. we cannot have the same value for x for which to have different results. Because then we do not know if we have f (1) how much will it be equal to. f (1), will it be 5, or will it be 6? We don’t know. Here we know that how much will be equal to f (1), to 7. Here we know how much f (2) will be equal to 7. So far so good. So when we have 2, we get 7. When the argument is 3, we get 8. If the argument is 4, we get 8. For example, let’s define the function so we get 8 if x is equal to 3 or 4. Then we have 5. The function is equal to 9 if x is equal to 5. For a subpoint, this is the definition of the function, fully compliant with the rules definition of the function. Whatever value you give me 1, 2, 3, 4, or 5, what values \u200b\u200bare actually the definition area in the case, I can tell you what will be the value of the function for each of these points. They will range around 7, 8, or 9. So, for a subpoint we definitely have a function. Now b point. Let’s see if x is 1 and y is 1. We are given that if x is 1, then y is minus 1. This makes no sense. Here is what they tell us here. They try to compose a function where we are told that this function will be equal to 1 if x is equal to 1, but or it will be equal to minus 1 if x is equal to 1. So when we have f (1), we don’t know how much will the function be equal to. Will it be equal to 1 or minus 1? We don’t know what to take, this here, or this here. Therefore, this is not a function. Sub-item b is not a function. This dependency is not a function. Okay. Let’s decide a little more. Task 6. Express each graph of a function by an equation. We have this letter-like thing. We can record it in several ways. Let’s call it f (x). We can also call it g (x), or h (x), but if we haven’t used it before that, we usually use f (x). This is x. Let’s see, we seem to have a straight line when x is greater than 0, and another straight line when x is less than 0. We have a case where x is greater than 0 number. And a separate case where x is less than 0. And I will try to connect the two cases in a second. What does this line look like to you here? When x is 0, y is 0. When x is 2, y is 1. When x is 4, y is 2. It seems that whatever number is x, y is always is 1/2 times less than x. When x is 6, y is 3. So it is equal to 1/2 when x is greater than 0. And when x is less than 0, when x is minus 2, ye 1. When x is minus 4, y is 2. It seems that this is exactly what 1/2 is about. Minus 1/2 to minus 4 is 2. We have 1/2 over x when x is less than 0. It seems like a completely correct answer. But if we want to make it a little easier, to clarify it a bit, we can write the function as f (x) is equal to something, instead of writing separately for greater than or less than 0, let’s just take the absolute value for x and multiply it by 1/2. Task 10.

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