Maybe you already know how to calculate a function, set with specific argument values. For example, if this table is the setting of our function, if someone says, “What is f (-9)?”, you can say that if we set as an argument -9 to our function, if x is -9, this table tells us that f (x) will be equal to 5. You may already have experience with complex (composite) functions, where you say f (f (-9) +1). This is interesting, it seems very difficult, but we know how much f (-9) will be – it will be 5 – so this will be f (5 + 1). This will be equal to f (6) and if we look in our table, f (6) is equal to -7. Everything so far is a negotiation, but I want us to start calculating the inverse functions. The function f is reversible, because it is a one-to-one comparison for all x and all f (x). No two hiccups are mapped to the same f (x), so this is a reversible feature. With that in mind, let’s see if we can calculate the inverse of the function of f (8). How much will that be? I encourage you to pause the video and try to think about it. As a negotiation of what functions do, f (x) will compare a value of this definition set (DM) of the corresponding value of the functional set (FM). This is the function f. This is the definition set (DM), and this is the functional set (FM). The inverse of the f function, if you take a value from FM, it will compare it back to the corresponding value in the definition set. But how to think about it this way? The inverse of f (8) function, this is the value that is compared to 8, that is, if it was 8, we would say, “What corresponds to 8?” We see that f (9) is 8, so the inverse of f (8) function will output – i will do it in the same color – will display 9. If that makes things easier, we can make a table. I will do this so as not to make a mistake somewhere. Where I will plot x and the inverse of f (x) function, and I will actually swap the places of these two columns. f (x) will display -9, with argument 5, the inverse of f (x) will output 5, with argument -9. I just swapped places with those two. We now compare this to this. So the inverse of f (x) will compare 7 to -7. Notice that instead of comparing this to this, we will now compare this to this. So the inverse of f will compare 13 to 5. Will match -7 to 6. Will match 8 to 9 and will also compare 12 to 11. I seem to have recorded them all. I just swapped places on these two columns. The inverse function of f maps this column to this column. I just swapped places with them. It’s getting a little clearer now. You see here, The inverse of f (8) function, if you give it as argument 8, you get 9. Now we can use this to do more interesting things. We can calculate something like f (f ^ (- 1) (7)). f ^ (- 1) is the notation of the inverse of f function (ed. note). Thus, f (f ^ (-1) (7)). How much will that be? Let us first calculate f ^ (- 1) (7). f ^ (-1) (7) compares 7 to -7. So it’s going to be f of that here, f ^ (- 1) (7), as you can see, is -7. And then to calculate the function, f (-7) is going to be 7. And this is logical. We compared f ^ (- 1) (7) to -7 and calculating the function from this, we went back to 7. Let’s take another example to understand all this by matching back and forth between these two sets, between the application of the function and the inverse function. Let’s calculate how much it will be equal to – i will do it in purple – how much will be equal to f ^ (- 1) (f ^ (- 1) (13)). How much will that be? I encourage you to stop the video and try to find it. What is the inverse function of f (13)? That, looking at this table here, f ^ (- 1), the inverse of f, compares 13 to 5. You see here that f compares 5 to 13, so the inverse f function will match 13 to 5. That is, f ^ (-1) (13) will be 5, so this is the same thing as f ^ (- 1) (5). And how much is f ^ (-1) (5)? f ^ (- 1) compares 5 to -9, so this will be equal to -9. Once again, f matches -9 to 5, so f ^ (- 1) compares 5 to -9. Initially, when you start solving these functions and inverse functions, it looks confusing because you’re moving back and forth, but you just have to remember that one function compares one set of numbers to another set of numbers. And the opposite of this function does the opposite. If the function matches 9 to 8, the inverse function will match 8 to 9. One way to think about this is to just swap the locations of these columns. I hope this clarifies more than confuses.

Algebra