Algebra

Domain and range of a function given a formula | Algebra II | Khan Academy

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“Define the definition set and the functional set of the function f (x) = 3x ^ 2 + 6x – 2. ” The definition set of the function is: what is the set of all valid input values or all valid x values \u200b\u200bfor this function? And I can take any real number, square it, multiply it by 3, then add 6 times this real number and then subtract 2 from it. Any number if we are talking about real numbers, when we are talking about an arbitrary number. The definition set, the set of valid input values, the set of input values \u200b\u200bfor which the function is set is all real numbers. The definition set here is all real numbers. You may be wondering if all numbers are real. You may or may not know that there is a class of numbers, which are a little weird when you first see them, called imaginary numbers and complex numbers. But I’m not going into that now. Most traditional numbers you know are part of the set of real numbers. It includes almost everyone except complex numbers. If you take any real number and put it here, you can square it, multiply it by 2, add 6 times that number and subtract 2. The functional set, at least as I think about it in this series of videos – the functional set is the set of possible output values \u200b\u200bof this function. Or if we say that y = f (x) of the graph, this is the set of all possible values \u200b\u200bof y. To realize this, I will try to draw a graph of this feature here. And if you know the square – such is this square function – you may already know that it has the form of a parabola. And her appearance may look like this. It has an opening up. But other parables have similar shapes. And you see, when a parabola has that shape, it will not accept values \u200b\u200bbelow its peak when open upwards, and will not accept values \u200b\u200babove its top when open down. Let’s see if we can draw this and find the top. There are ways to accurately calculate the peak, but let’s see how we can think about this task. I will try some values \u200b\u200bof x and y. There are other ways to directly calculate the peak. The formula for it is -b / 2a. Derived directly from the formula for finding solutions of a quadratic equation, which you get when you complete the square. Let’s test some values \u200b\u200bof x and see how much f (x) is. These are the values \u200b\u200bwe tested in the last two videos. What happens when x = -2? Then f (x) is 3 over (–2) ^ 2, which is 4, plus 6 * (- 2), which is -12, minus 2. That’s 12 – 12 – 2. This is equal to -2. What happens when x = -1? This is going to be 3 over (-1) ^ 2, which is just 1, minus, or I must say plus, 6 * (- 1), which is -6, and after minus 2. That’s 3 – 6 is -3, minus 2 is equal to 5 and this is actually the top. You know the formula for the tip, I repeat it again, it is -b / 2a. –B. This is the coefficient of this article here. That’s -6/2 on that here, 2 * 3. 2 * 3 and all this is equal to -1. This is the top, but let’s move on. What happens when x = 0? The first two terms are 0 and you only have -2 left. When x = +1 … Here you can see that this is the tip and you start to see the symmetry. If you go 1 over the top, f (x) = –2. If you go by an x \u200b\u200bvalue below the vertex or below the value of x at the vertex, f (x) is again equal to -2. But let’s move on. Let’s make another point here. x = 1. When x = 1 you have 3 over 1 ^ 2, which is just 1. 3 * 1 plus 6 * 1, which is 6, minus 2. That’s 9 – 2 and it’s equal to 7. And I think those are enough points, to give us an idea of \u200b\u200bwhat this graphic will look like. What the graph of the function will look like. It will look like this. I will draw it as best I can. This is x = -2. We draw the whole axis. This is x = -1, this is x = 0 and this is x = 1. Then when x is equal to … we go from -2 all the way to the positive part. Or we have to go from -5 all the way to +7. Let’s say these are the negative 1, 2, 3, 4, 5. This here on the y-axis is -5, and then we will move to +7. 1, 2, 3, 4, 5, 6, 7. I can go on, this is in the direction of y and we will set y to be equal to the output value of the function. y = f (x). And this here is 1. Let’s put the dots. You have the point (-2; -2). When x is -2, this is the x-axis. When x is -2, y is -2. y is -2, and this is exactly after 3. This is the point (-2; -2). Okay. Then we have this point, which is purple. When x is -1, f (x) is -5. And we have already said that this is the top. After a while you will see the symmetry around it. This is the point (-1; -5). And then the point (0; -2). When x is 0, y is -2 for f (x) = –2 or f (0) = –2, so this is the point (0; –2). And then, finally, when x is 1, f (1) is 7. This here is the point (1; 7) and it gives us an idea of \u200b\u200bwhat this parabola, this curve, will look like. I will draw it as best I can. It will look like this. And it will continue in that direction. It will continue in this direction. But I think you see the symmetry around the top. And if you have to … If you have to put rights here, both sides are a kind of mirror images of each other. You can turn them over, and that’s the way we understand that this is the tip. And so we also understand, because this is an open up parabola. There are formulas for the vertex and there are many ways to calculate it. But since it’s a parabola with an opening up, then the peak will be the minimum point. This is the minimum value that the parabola will accept. And let’s go back to the original question, which is to try to find the functional set, the set of y values, the set of output values \u200b\u200bthat this function can generate. You can see that the function can go down to -5. It goes down to -5 at the top. But as you go right, while the values \u200b\u200bof x increase to the right or decrease to the left, the parabola goes up. So the parabola can’t give you values \u200b\u200b- f (x) will never be less than –5. But our definition set can accept all values. It can continue to increase forever, until x becomes larger or smaller and moves away from the top. Our functional set – we have already said that our definition set is all real numbers. Our functional set, the possible values \u200b\u200bof y, are all real numbers greater than or equal to -5. It can accept the value of any real number greater than or equal to –5. Nothing less than -5.

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