You need reasoning similar to that used here.
Hello! My question is as follows:
f(x) = (x-3)^3 , x > 3
Find f^-1(f(x)) and f(f^-1(x)), stating clearly the rules and domains.
I know that the rules of both are the same: f^-1(f(x)) = x and f(f^-1(x)) = x. However, I don't understand why are the domains different? Please explain to me! Thank you!
I understand why f^-1(f(x)) has a domain of x>3, but I don't know why the domain of f(f^-1(x)) is x > 0. Is it because since f(x) has a domain of x >3, so the range of f inverse is also [3, infinity). f inverse = (cube root x) + 3. For the range of f inverse to be [3, infinity), x > 0? Is my reasoning correct?
There is a rule that says:
These rules can be thought of as a consequence of swapping y and x around in y = f(x) to get the rule for the inverse function ..... y's and x's swap --> domain and range swap .....
In your example,
and .
Therefore:
.
.
The composite function g(f(x)) exists iff . When that condition is met, the domain of g(f(x)) is . In you present question, this condition is met for both and .
So:
The domain of is .
The domain of is .