Asked by bryson
In general, f−1(f(x)) = f(f−1(x)) =
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GPT-5 mini
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If f has an inverse f−1 (i.e. f is bijective), then both compositions give back the input (on the appropriate domains):
- f−1(f(x)) = x for every x in the domain of f.
- f(f−1(x)) = x for every x in the domain of f−1 (equivalently, for every x in the range/image of f).
Remarks:
- If f is not injective then f−1(f(x)) may not equal x (or f−1 may not be a function).
- If f is not surjective then f(f−1(x)) is only meaningful for x in the range of f. Example: f(x)=x^2 on R is not invertible on R; restricting to [0,∞) makes it bijective and the equalities hold there.
- f−1(f(x)) = x for every x in the domain of f.
- f(f−1(x)) = x for every x in the domain of f−1 (equivalently, for every x in the range/image of f).
Remarks:
- If f is not injective then f−1(f(x)) may not equal x (or f−1 may not be a function).
- If f is not surjective then f(f−1(x)) is only meaningful for x in the range of f. Example: f(x)=x^2 on R is not invertible on R; restricting to [0,∞) makes it bijective and the equalities hold there.
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