Example where an inverse function does not equal the elements

Question

I'm trying to find an example of a function $f: A to B$ and $X subset A$ so that $f^{-1}(f(X)) ne X$, and similarly where $Y subset B$ so that $f(f^{-1}(Y)) ne Y$.
I thought to have $f = x^2$, which has no inverse, thus making it vacuously true that $f^{-1}(f(X)) ne X$ and $f(f^{-1}(Y)) ne Y$, but that seems like a copout. Any help is greatly appreciated.

TheSilverDoe · Accepted Answer

Consider $f : mathbb{R} rightarrow mathbb{R}$ defined for all $x in mathbb{R}$ by $f(x)=0$, and $X = lbrace 0 rbrace$ and $Y = mathbb{R}$.

Shubham Johri · Answer

Here $f^{-1}$ does not denote inverse function of $f$ but inverse image of a set, which is defined even when $f$ is not invertible. In fact your search for an invertible example will be in vain as for invertible $f,f(f^{-1}(Y))=Y$ and $f^{-1}(f(X))=X$ for all $Xsubseteq A,Ysubseteq B$.
$y(x):Bbb RtoBbb R,y=x^2$ is a valid example. Take $X={1}implies f^{-1}(f(X))=f^{-1}({1})={pm1}$.
Take $Y={0,-1}$. Then $f(f^{-1}(Y))=f({0})={0}$.

Example where an inverse function does not equal the elements

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