MHB Why is the following not true: f(f^(-1)(B))=B?

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The equation f(f^(-1)(B))=B is not universally true because it depends on the properties of the function f. If f is not surjective, meaning its range does not cover the entire set B, then f(f^(-1)(B)) will equal f(A) instead of B. This demonstrates that the equality fails when the function does not map onto all elements of B. Thus, the condition of surjectivity is crucial for the equation to hold true. Understanding the implications of function properties is essential in evaluating such expressions.
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The title says it: why is the following not true: $f(f^{-1}(B))=B$?

Thanks!
 
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Re: Why is the following not true: $f(f^{-1}(B))=B$?

OhMyMarkov said:
The title says it: why is the following not true: $f(f^{-1}(B))=B$?

Thanks!
You have not given a context for this, but in general if you have a function $f:A\to B$ there is no reason to suppose that the range of $f$ is the whole of $B$. If $f$ is not surjective then $f(f^{-1}(B)) = f(A) \ne B$.
 

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