If $f:\{1,2,3\}\to\{1,2,3\}$ is bijective and $f(1)=2$, can we verify that $f(2)=1$?

Question

I know that if $f$ is bijective, if $f(1)=2$ then $f^{-1}(2)=1$ but if $f:\{1,2,3\}\rightarrow\{1,2,3\}$ then does it mean necessarily that $f(2)=1$?

Answer 1

No, in general just because $f(1) = 2$, we don't have $f^{-1}(2) = 1$. For example, let $$ f(1) = 2, f(2) = 2, f(3) = 1. $$ Then $f^{-1}(2) = \{1,2\}$. More properly we would write $f^{-1}(\{2\}) = \{1,2\}$.
Generally, for any set $A \subseteq X$, we have $$ A \subseteq f^{-1}(f(A)), $$ and equality when $f$ is injective.