Independent and Dependent Variables
This video explains the differences between dependent and independent variables and inverting functions.
Dependent and Independent Variables and Inverting
Before we move on to more complicated functions, I want to spend a little more time talking about some of the properties of functions. I mentioned briefly in the last video that a function could be thought of, essentially, as a two-variable equation. For instance, the equation y equals 2x, plus 1 is essentially the same thing as f of x equals 2x, plus 1.
These are both equations of two variables, where one variable is solved in terms of the other variable. One of the main differences, though, is that a function, f of x, immediately defines what your independent variable is. The x in the function, f of x on this side tells us that x is our independent variable and that changes in x will bring about changes in f of x, whereas with this function we could just as easily have written it as x in terms of y, so there’s the advantage of immediately knowing which is your independent variable and which is your dependent variable, so that’s one thing.
One additional property of functions that’s helpful is that if you have a function such as this one, but your input variable changes, or your input variable is scaled differently or is offset, you can just plug it into the same function and have a new output.
For instance, let’s say we use this function to measure something, x, but now the value of our input has changed from x to 3x plus 2, so now where we used to have x, we have 3x plus 2. If we want to keep the same function notation we can write now, f of 3x plus 2 is equal to 2, times 3x plus 2, plus 1, or 6x plus 4, plus 1, or 6x plus 5, so f of 3x plus 2 equals 6x plus 5, where this is the original function.
You have some tricks you can do to maintain the same equation while changing the input variable and changing the input variable is going to come into play when you have shifts in graphs. Let’s look for a second now at graphing of functions. The graphing of a function is essentially the same as the graphing of a linear equation, or any equation, really.
We have our x variable and our f of x variable and we can plot them like this (so go ahead and mark off the locations of the numbers there). Let’s plot the line f of x equals 2x, plus 1. When x equals 0, f of x equals 1, so we can write a point of the line right here.
When x equals 1, f of x equals 3, so we can write another point here, and from these two points we can extrapolate that the line looks something like that, so this is the graph of our function f of x equals 2x, plus 1. Now suppose that we want to shift this line two units to the right.
If we want to shift this line two units to the right, if we were working with just this sort of equation up here, we would just have to calculate a whole new equation from scratch based on the properties of the new line. With the functional notation, all we have to do is change this—we’re saying we’re moving the function two places to the right, so that means the value of x is now going to be two greater than it used to be—so we can rewrite this as f of x, minus 2 and that will give us 2, times x minus 2, plus 1, or 2x minus 4, plus 1, or minus 3, so 2x minus 3.
We can go ahead and graph this one, it’s going to come through down here at (minus 3, 0), then at (1, negative 1), and at (2, 1), so a new line is going to be over here which, if you’ll notice, is two units to the right of where it used to be. By having this in function notation we can easily plug in the change in our variable, either the input variable or, if we wanted to shift the graph vertically, we could put a change in the output variable to mirror the change in the line.
When you’re moving graphs to the right, you’re going to subtract from the x whatever value you’re moving it to the right; if you want to move it to the left, you would add that value however far you’re shifting it. To shift It up, you would add to the very end of your equation, so if we wanted to shift the graph up by 2, we would add 2 to the end of, the right of, the equation; if we were going to move it down 2, we would subtract 2.
The functional notation here allows us to make shifts to the graph a lot more easily and cleanly than we could otherwise. The last thing I want to look at for this video is invertibility. Some functions, as I mentioned in the last video, some functions can have an input of 1, and an output of 2, and an input of 2, and an output of 2, so they can have two different inputs that both go to the same number, and this is fine for a function.
However, if a function has multiple inputs that go to the same output, the function is not invertible. When you invert a function, essentially what you’re doing is you’re switching the range and the domain, (so the range becomes the new domain, and the domain becomes the new range) and the reason that a situation like this is not invertible is because when you switch the two you have, for instance, 2 going to a 1, and then 2 going to a 2, and this violates the rules of a function.
This case here is not representative of an invertible function. A way to look at it graphically is if you can draw vertical lines anywhere in the function and cross the function more than once, it’s not actually a function in that case—so if you draw vertical lines and cross some figure more than once, it’s not a function.
If you cross a function with horizontal lines—or if you draw horizontal lines across a function—and you can cross the function more than once, then it’s not an invertible function. Let’s say, for instance, that we have a function that looks like this (it comes up and then it goes back down).
It’s a function because we can draw horizontal lines all through it and only intersect it once on each line, But if we start drawing horizontal lines through it, you see we intersect it in two points here, so that would disqualify it from being invertible. Any linear function is going to be invertible because the straight line—any horizontal line is only going to cross it once, any vertical lines only going to cross once—so lines are invertible, and there are a few other functions that are invertible as well, but not all functions will be invertible.