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### Course: College Algebra>Unit 11

Lesson 4: 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.
• 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)
• why u didnt use that definition of composition