Directly Proportional and Inversely Proportional
In mathematics, two variables are proportional if there is always a constant ratio between them. (for details see Ratio). The statement "y is inversely proportional to x" is written mathematically as "y = c/x." This is or constant of proportionality. This can also be viewed as a two-variable linear equation with a y-intercept of 0. Learn about and revise ratio, proportion and rates of change with this BBC Equations involving inverse proportions can be used to calculate other values. In mathematics, an inverse function (or anti-function) is a function that "reverses" another Not all functions have inverse functions. Stated otherwise, a function, considered as a binary relation, has an inverse if This property is satisfied by definition if Y is the image (range) of f, but may not hold in a more general context .
So if x is equal to 1, then y is 2 times 1, or is 2. If x is equal to 2, then y is 2 times 2, which is going to be equal to 4.
Direct and inverse proportion
So when we doubled x, when we went from 1 to so we doubled x-- the same thing happened to y. So that's what it means when something varies directly. If we scale x up by a certain amount, we're going to scale up y by the same amount. If we scale down x by some amount, we would scale down y by the same amount.
And just to show you it works with all of these, let's try the situation with y is equal to negative 2x. I'll do it in magenta.
Let's try y is equal to negative 3x. So once again, let me do my x and my y. When x is equal to 1, y is equal to negative 3 times 1, which is negative 3.
Intro to direct & inverse variation (video) | Khan Academy
When x is equal to 2, so negative 3 times 2 is negative 6. So notice, we multiplied. So if we scaled-- let me do that in that same green color.
If we scale up x by it's a different green color, but it serves the purpose-- we're also scaling up y by 2. To go from 1 to 2, you multiply it by 2. To go from negative 3 to negative 6, you're also multiplying by 2.
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So we grew by the same scaling factor. To go from negative 3 to negative 1, we also divide by 3. We also scale down by a factor of 3. So whatever direction you scale x in, you're going to have the same scaling direction as y.
That's what it means to vary directly. Now, it's not always so clear. Sometimes it will be obfuscated. So let's take this example right over here. And I'm saving this real estate for inverse variation in a second. You could write it like this, or you could algebraically manipulate it. Or maybe you divide both sides by x, and then you divide both sides by y. These three statements, these three equations, are all saying the same thing.
So sometimes the direct variation isn't quite in your face.
But if you do this, what I did right here with any of these, you will get the exact same result. Or you could just try to manipulate it back to this form over here. And there's other ways we could do it. We could divide both sides of this equation by negative 3. And now, this is kind of an interesting case here because here, this is x varies directly with y.
Or we could say x is equal to some k times y. And in general, that's true. If y varies directly with x, then we can also say that x varies directly with y. It's not going to be the same constant. It's going to be essentially the inverse of that constant, but they're still directly varying. Now with that said, so much said, about direct variation, let's explore inverse variation a little bit. Inverse variation-- the general form, if we use the same variables. And it always doesn't have to be y and x.
It could be an a and a b. It could be a m and an n. If I said m varies directly with n, we would say m is equal to some constant times n. Now let's do inverse variation. So let me draw you a bunch of examples. And let's explore this, the inverse variation, the same way that we explored the direct variation. And let me do that same table over here.
So I have my table. I have my x values and my y values. If x is 2, then 2 divided by 2 is 1.
Proportionality (mathematics) - Wikipedia
So if you multiply x by 2, if you scale it up by a factor of 2, what happens to y? Most of the steps are not all that bad but as mentioned in the process there are a couple of steps that we really need to be careful with since it is easy to make mistakes in those steps. For all the functions that we are going to be looking at in this course if one is true then the other will also be true.
However, there are functions they are beyond the scope of this course however for which it is possible for only one of these to be true. This is brought up because in all the problems here we will be just checking one of them.
We just need to always remember that technically we should check both. However, it would be nice to actually start with this since we know what we should get. This will work as a nice verification of the process. Here are the first few steps.
The next example can be a little messy so be careful with the work here. With this kind of problem it is very easy to make a mistake here. That was a lot of work, but it all worked out in the end. We did all of our work correctly and we do in fact have the inverse. There is one final topic that we need to address quickly before we leave this section.
There is an interesting relationship between the graph of a function and the graph of its inverse. Here is the graph of the function and inverse from the first two examples. This will always be the case with the graphs of a function and its inverse.