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Algebra II (2018 edition)

Unit 1: Lesson 8

Verifying that functions are inverses

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!

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• 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, by definition inverse functions will always be inverses the other way.
Note that f(x) = x^2 is not truly invertible because its inverse is technically not a function.
• Is it always true that if either f(g(x))=x OR g(f(x))=x, then it must also be the case that f(g(x))=g(f(x))=x and the functions are inverses?
• If f and g both have all of ℝ as both their domains and ranges, then yes. But if the domains or ranges are restricted, then not necessarily.

For example, sin(x) has domain ℝ and range [-1, 1]. arcsin(x) has domain [-1, 1] and range [-π/2, π/2].

So if we take a number outside of those ranges, like 3π/4, we have that arcsin(sin(3π/4))=π/4, not 3π/4.
• 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?
• As one gets more comfortable with these types of problems, is it okay to use your intuition? I know that simplifying the compounded functions is the only way to make sure you're correct but sometimes you can tell that things are going to cancel out without even having to do the problem. Also, if, say you're given f(x) and h(x), and f(h(x)) simplifies to x, is it reasonably safe to assume that h(f(x)) will equal x too?
• I quote @kubleeka : 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).
• Can someone help me find the inverse of the following function? I've asked some of my friends and they cant figure it out either. F(x)=x^2+3/5
• ``sqrt(x-(3/5))``
(1 vote)
• So basically what Sal is saying at is if f(g(x)) is equal to g(f(x)), they're inverse functions, right?
• Correct. If they are not equal, then the functions are not inverses.
(1 vote)
• 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).
• Are every composite functions inverses of each other?
(1 vote)
• No. You can compose any two functions you like, so long as their domains and ranges match up right. That is, so long as the outer function is defined for whatever the inner function gives.
• 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.