If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: Precalculus>Unit 1

Lesson 5: Verifying inverse functions by composition

# Verifying inverse functions by composition: not inverse

Sal composes f(x)=2x-3 and g(x)=½x+3, and finds that f(g(x)) ≠ g(f(x)) ≠ x, which means the functions are not inverses.

## Want to join the conversation?

• Is it possible to have two functions such that f(x) = g^-1(x), but g(x) does not = f^-1(x)?
• That's a really good question. I think if one function equals the inverse of the other function then it must work vice versa.
• can you have one function = x but not the other?
• No, if either f(g(x)) = x or g(f(x)) = x then they have to be inverse functions, since the functions cancel each other out to result in the unchanged x.
• What if f(g(x)) = x + 3 but g(f(x)) = x + 3? Would they still be equal? How about if one was x + 3 and the other was x -3? How would that be different?
• They have to both equal x to be inverse.
• How did this happen?
f(g(x))=3(1/3x^2-9)^2+9
=3(1/9x^4-6x^2+81)+9
This is from Khan Academy (in the hint).
From where they brought (-6x^2)?
(1 vote)
• I've got you:
the x^2 in f(x) is replaced with the function g(x)* which is *(1/3x^2-9)*, which makes *(1/3x^2-9)*^_2_, which can be expanded as *(1/3x^2-9)(1/3x^2-9)*, which equals *1/9x^4-9/3x^2-9/3^2+81. Add the two like terms, *(-9/3x^2-9/3x^2)* and you get *(-18/3x^2)*, which is simplified as *(-6x^2)*, which gives you 1/9x^4-6x^2+81, then you can solve for the rest of f(g(x)), then do the same for g(f(x)).

To simplify,
f(g(x)) = 3(1/3x^2-9)^2+9
multiply (1/3x^2-9)(1/3x^2-9)
simplify (1/9x^4-9/3x^2-9/3^2+81)
f(g(x))=3(1/9x^4-6x^2+81)+9
distribute (3[1/9x^4]) (3[-6x^2]) (3[81]) + 9
f(g(x)) = 1/3x^4-18x^2+252

I hope that helps!
• Must both my answers for the composited inverse functions equal just X? Or if they both equaled x+2, would they still be inverse functions of each other.
• If an function is an inverse of another, it takes the output value of the original function as its input and returns the input value of the original function. Basically, one function reverses all steps performed by the other function. Thus, when you do the composite function check to see if functions are inverses of each other, the final result must be "x". If you are getting "x+2", then the original input value has changed. So, the functions are no inverses.

Hope this helps.
• If in the place of x there may be ( x+a) or (x-b) such that f(g(x))=g(f(x)), then are they considered as inverse functions?
• As long as both functions give the same answer, yes.
• Hi
If we work out that f(g(x)) = 0, does that automatically mean that g(f(x)) = 0?
• If 𝑓 and 𝑔 are inverses, then the answer is always yes. Because:
𝑓(𝑔(𝑥)) = 𝑔(𝑓(𝑥)) = 𝑥
So in your case, if 𝑓 and 𝑔 were inverses, then yes it would be possible. (This also implies that 𝑥 = 0).
However, if 𝑓 and 𝑔 are arbitrary functions, then this is not necessarily true. One can easily construct a counter example. Try to do so yourself! Comment if you have questions!
• Is it possible that f(g(x)) = x but g(f(x)) is not equal to X.. I mean that why do we prove that both the equations have a final value of x in order to show that both the equations are inverses of each other. ?