Functions:Inverse Image

The inverse image (or preimage) of a given subset $B$ of the codomain of $f,$ is the set of all elements of the domain that map to the members of $B.$

Inverse image

Let $f$ be a function from $X$ to $Y.$ The preimage or inverse image of a set $B\subseteq Y$ under $f,$ denoted by $f^{-1}[B],$ is the subset of $X$ defined by

$$f^{-1}[B]=\{x\in X\,|\,f(x)\in B\}$$

.

Other notations include $f^{-1}(B)$ and $f^{-}(B)$. The inverse image of a singleton set, denoted by $f^{-1}[\{y\}]$ or by $f^{-1}[y],$ is also called the fiber or fiber over $y$ or the level set of $y$. The set of all the fibers over the elements of $Y$ is a family of sets indexed by $Y$.

For example, for the function $f(x)=x^{2},$ the inverse image of $\{4\}$ would be $\{-2,2\}$. Again, if there is no risk of confusion, $f^{-1}[B]$ can be denoted by $f^{-1}(B),$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}} can also be thought of as a function from the power set of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} to the power set of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X.} The notation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}} should not be confused with that for inverse function, although it coincides with the usual one for bijections in that the inverse image of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} under Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is the image of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} under Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}}