Inputs & outputs of inverse functions

You may by now be familiar with the notion of evaluating a function with a particular value, so for example, if this table is our function definition, if someone were to say, "Well, what is f of -9?" you could say, okay, if we input -9 into our function, if x is -9, this table tell us that f of x is going to be equal to 5. You might already have experience with doing composite functions, where you say, f of f of -9 plus 1. So this is interesting, it seems very daunting, but you say, well we know what f of -9 is, this is going to be 5, so it's going to be f of 5 plus 1. So this is going to be equal to f of 6, and if we look at our table, f of 6 is equal to -7. So all of that is review so far, but what I want to now do is start evaluating the inverse of functions. This function f is invertable, because it's a one-to-one mapping between the xs and the f of xs. No two xs map to the same f of x, so this is an invertable function. With that in mind, let's see if we can evaluate something like f inverse of 8. What is that going to be? I encourage you to pause the video and try to think about it. So f of x, just as a reminder of what functions do, f of x is going to map from this domain, from a value in its domain to a corresponding value in the range. So this is what f does, this is domain... and this right over here is the range. Now f inverse, if you pass it, the value and the range, it'll map it back to the corresponding value in the domain. But how do we think about it like this? Well, f inverse of 8, this is whatever maps to 8, so if this was 8, we'd have to say, well, what mapped to 8? We see here f of 9 is 8, so f inverse of 8 is going to be equal to 9. If it makes it easier, we could construct a table, where I could say x and f inverse of x, and what I'd do is swap these two columns. f of x goes from -9 to 5, f inverse of x goes from 5 to -9. All I did was swap these two. Now we're mapping from this to that. So f inverse of x is going to map from 7 to -7. Notice, instead of mapping from this thing to that thing, we're now going to map from that thing to this thing. So f inverse is going to map from 13 to 5. It's going to map from -7 to 6. It's going to map from 8 to 9, and it's going to map from 12 to 11. Looks like I got all of them, yep. So all I did was swap these columns. The f inverse maps from this column to that column. So I just swapped them out. Now it becomes a little clearer. You see it right here, f inverse of 8, if you input 8 into f inverse, you get 9. Now we can use that to start doing fancier things. We can evaluate something like f of f inverse of 7. f of f inverse of 7. What is this going to be? Let's first evaluate f inverse of 7. f inverse of 7 maps from 7 to -7. So this is going to be f of this stuff in here, f inverse of 7, you see, is -7. And then to evaluate the function, f of -7 is going to be 7. And that makes complete sense. We mapped from f inverse of 7 to -7 and evaluating the function of that, went back to 7. So let's do one more of these just to really feel comfortable with mapping back-and-forth between these two sets, between applying the function and the inverse of the function. Let's evaluate f inverse of f inverse of 13. f inverse of 13. What is that going to be? I encourage you to pause the video and try to figure it out. What's f inverse of 13? That's, looking at this table right here, f inverse goes from 13 to 5. You see it over here, f went from 5 to 13, so f inverse is going to go from 13 to 5. So, f inverse of 13 is going to be 5, so this is the same thing as f inverse of 5. And f inverse of 5? -9. So this is going to be equal to -9. Once again, f inverse goes from 5 to -9. So at first when you start doing these functions and inverse of functions it looks a little confusing, hey, I'm going back and forth, but you just have to remember a function maps from one set of numbers to another set of numbers. The inverse of that function goes the other way. If the function goes from 9 to 8, the inverse is going to go from 8 to 9. So one way to think about it is, you just switch these columns. Hopefully, that clarifies more things than it confuses.