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### Course: Precalculus>Unit 1

Lesson 5: Verifying inverse functions by composition

# Verifying inverse functions from tables

Inverse functions undo each other. Functions s and t are inverses if and only if s(t(x))=x and t(s(x))=x for every x-value in the domains. Created by Sal Khan.

## Want to join the conversation?

• first! first time I've seen a Khan video without any comments here. Awesome video! Keep it up!
• Is it just me, or is the video a bit quiet?
• Sometimes, the audio volumes of certain videos aren't similar to other videos because they may have been uploaded at different days, with different audio settings.
• Is there a relationship between the composite function s(t(x)) having all of its values equal and function s(x) and function t(x) being inversible or is it just a coincidence in this particular problem?
• If the composite function s(t(x)) has all of its values equal, this implies that the function t(x) maps all inputs to the same output, which means that t(x) is a constant function. Let's call this constant output "c".

If t(x) is a constant function with output "c", then the composite function s(t(x)) simplifies to s(c) for all inputs of x. Therefore, if s(t(x)) has all of its values equal, then s(c) must also have a constant value.

If s(c) has a constant value, this implies that the function s(x) must also be a constant function, since it maps all inputs to the same output.

In summary, if the composite function s(t(x)) has all of its values equal, then both function s(x) and function t(x) must be constant functions, and in this case, they are trivially inverses of each other. However, it should be noted that this is a special case and does not necessarily apply to all situations where the composite function has all of its values equal.
• Here we have `s(x) = t^-1(x)`.

Does this mean `t(x) = s^-1(x)`?
• Yes, assuming both functions are invertible to begin with. Taking the inverse of an inverse function will yield the original function.
• why u didnt use that definition of composition
(1 vote)
• If the tables didn't give all of the function input-output pairs, would the function not have been inverses of each other?
(1 vote)
• If a function does not give all of the input-output pairs for both functions, then it is possible that the functions may not be inverses of each other. In order for two functions to be inverses of each other, they must satisfy the following criteria:

1. Each input of one function corresponds to a unique output of the other function.
2. The composition of the two functions in either order results in the identity function.

If the tables do not give all of the input-output pairs for both functions, it may be impossible to determine if the functions satisfy these criteria, and thus whether they are inverses or not. In such cases, additional information or data would be needed to determine if the functions are inverses of each other.
(1 vote)

## Video transcript

- [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.