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### Course: Class 12 math (India)>Unit 1

Lesson 5: Verifying inverse functions by composition

# Verifying inverse functions by composition

Sal composes f(x)=(x+7)³-1 and g(x)=∛(x+1)-7, and finds that f(g(x))=g(f(x))=x, which means the functions are inverses!

## Want to join the conversation?

• Would it be possible to have two functions where f(g(x)) maps back to x but g(f(x)) doesn't? In other words: if two functions are inverses one way, will they always be inverses the other way?
• Yes a function and its inverse are both inverses of each other.
• So basically what Sal is saying at is if f(g(x)) is equal to g(f(x)), they're inverse functions, right?
• Well, the point isn't that their equal. The main point is that they take us back to x. That is considered the main reason (At least to me).
• Is this true for all functions?
• The method shown in the video is a common way to check if two functions are inverses of each other. If
f(g(x)) = x and
g(f(x)) = x for all
x in the domain of the functions, then
f(x) and
g(x) are inverses of each other. If this isn't true, then they're not inverses. Just find out if f(g(x)) is equal to g(f(x)). This is a handy tool for proving or disproving if functions are inverses!
• Do we know that g(x) is the inverse of f(x) because f(g(x)) and g(f(x)) equal x, or is it because f(g(x)) = g(f(x))?

For example, if it were the first, whenever we solve for a function of a function and get just x, that would mean they are inverses.

However, if it were the second, we would have to solve both ways to compare answers to make sure its the same.
• They are inverses of each other because f(g(x)) = x = g(f(x)). Or looking at it another way, g(x) = f^-1(x) "g undoes f". Or f(x) = g^-1(x) and "f undoes g". I suspect that you only have to solve it one way, find you got x and there is your inverse.
(1 vote)
• I know that this seems pretty obvious, but ¿Is always a function the inverse of its inverse?
• yes it is, but only if the inverse exists
• How do I solve for this video's set of equations?
I know the two functions and graphed them on desmos.
https://www.desmos.com/calculator/8eunthihxq
The solution for this set of functions appears to be (-9,-9) but how do I show this algebraically?
• If I am verifying inverse functions by compostion and I do f(g(x)) and get x as a result, do I also need to do g(f(x))?
• If you know that f has an inverse (nevermind what it is), and you see that f(g(x))=x, then apply f ⁻¹ to both sides to get
f ⁻¹(f(g(x))=f ⁻¹(x)
g(x)=f ⁻¹(x)

So if you know one function to be invertible, it's not necessary to check both f(g(x)) and g(f(x)). Showing just one proves that f and g are inverses.

You know a function is invertible if it doesn't hit the same value twice (e.g. if the functions is strictly increasing or decreasing).
• but g(-2) is non real so why is f(g(-2))=-2?
(1 vote)
• If you are referring to the functions in the video, g(-2) is a real number. You can do cube roots of negative numbers. g(-2) = ∛(-2+1)-7 = ∛(-1)-7 = -1-7 = -8.
Notice: ∛(-1) = -1 because (-1)^3 = -1

A square root of a negative number is not a real number. But, cube roots of negative numbers are real numbers.

Hope this helps.
• Do you always have to check 2 cases? I mean f(g(x)) must be equal to x and g(f(x)) must be equal to x? Or it's enough to check only one composite function?
(1 vote)
• There is no need to check the functions both ways. If you think about it in terms of the function f(x) "mapping" to the result y_ and the inverse f^-1(x) "mapping" back to _x in the opposite direction, one always gives you the result of the other. Therefore, once you have proven the functions to be inverses one way, there is no way that they could not be inverses the other way.

Similarly, if you look at a graph, inverses will always mirror each other over the line y=x. There is no way you can alter a reflection to let it be true one way and not the other!