Image $f(A)$ and Preimage $f^{-1}(B)$

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Definition (Image $f(A)$ and Preimage $f^{-1}(B)$)

Consider the function $f:X\to Y$. Let $A$ be a subset of the domain $X$ and let $B$ be a subset of the codomain $Y$.

• The image of $A$ under $f$ is the subset of $Y$ defined by $$ \begin{align} f(A) &=\{y\in Y\mid y=f(a) \text{ for some }a\in A\}\\ &=\{f(a)\mid a\in A\} .\end{align}$$ This means that $y\in f(A)$ if and only if $y=f(a)$ for some $a\in A$.
• The preimage (or inverse image) of $B$ under $f$ is the subset of $X$ defined by $$ \begin{align} f^{-1}(B) &=\{x\in X\mid f(x)=b \text{ for some }b\in B\}\\ &=\{x\in X\mid f(x) \in B\} .\end{align}$$ This means that $x\in f^{-1}(B)$ if and only if $f(x)=b$ for some $b\in B$ if and only if $f(x)\in B$. Note that when the set $B$ contains a single element, then we write $f^{-1}(b)$ rather than $f^{-1}(\{b\})$.