Verifying inverse functions by composition
Current time:0:00Total duration:4:32
- [Instructor] We're told the following tables, give all of the input output pairs for the functions S and T. So we see this first table here, we have some Xs and then they tell us what the corresponding S of X is. And then in this table that we have some Xs and they tell us the corresponding T of X. It says complete the table for the composite function, S of T of X, we wanna fill in these five entries here. And then they ask us our S and T inverses. So pause this video and see if you can figure this out on your own before we work through it together. All right, now let's work through this together. So let's just remind ourselves what's going on with a composite function like this. So you're going to take some X value and it looks like we're first going to put it into the function T, that is going to output T of X. And then we're gonna take that output, take that T of X, and then it will be the input into S. So then we're gonna input that into S and then that would output S of what we inputted which in this case is T of X. So let's go on that journey. So what we're gonna do is first take these numbers, put them into the function, T figure out what it outputs and then take that output and then put it into the function S, it's going to be a fun little ride. All right, so when X is equal to 12, we're gonna put it into our function T first. So when X is an input into T the output is equal to negative one. So that's our T of X, and there we're take this negative one and input it into S, so negative one here. And when you input that into S you get as the output, S of negative one is 12, so S of T of X is 12. So interestingly, this is 12. Now let's do the next one. So when we input 18 into T, so the 18 is the input. The T of X, T of 18 is two. And then if we wanna do, if we wanna input that into S, so this is gonna be the input into S, the output is 18. Very interesting, alright, let's keep going. So when we input 61 into T, the output is eight. Then when we take eight and we input it into S of X or S of eight I should say, is going to be 61. All right, things are looking good so far and I'm running out of colors, I'll do green. So when we take 70 and we input it into T, T of 7O is seven, when you take seven and input it into S you get 70, all right. And then one last one I will do in this blue color. When you take 100, input it into T it outputs negative five you take negative five input into S you get 100. So in every situation that we have looked at right over here, in all of these situations, we see that S of T of X is equal to X, which inclines us to believe that they are inverses. Remember if these two are inverses of each other, this would be true and also T of S of X is going to be equal to X. But we don't really know 100% unless we know that we have looked at every combination in the domains for each of them. Now, when you look at these, the two tables up here and I could have done this, this is the one we looked at on our journey to get to this 12 right over here. The following table gives all of the input output pairs for the function S and T. So this right over here is the domain for the function S and this right over here is the domain for the function T. So because for every member of the function S, every member of the domain of the function S, the corresponding output right over there is the domain for the function T and it takes us back to where we began. And then the opposite is true as well, for every member of the domain of T, what it outputs, that is the all of the possible inputs for X and they all take us back to where we began. So yes, the functions are inverses.