pre-image

Preimage / inverse image

For a function $f:A\to B$ and a subset $C\subseteq B$, the preimage or inverse image of $C$ under $f$ is defined as:

$$f^{-1}(C):=\{x\in A:f(x)\in C\}$$

is the set of all elements in the domain $A$ that get mapped to elements in $C$ under $f$.
Note that this notation does not require $f$ to be invertible – that would be $f(\cdot {)}^{-1}$ – it just describes which inputs lead to outputs in $C$.

Example

For $f(x)=x^2$ and $C=[1,4]$, we have:

$$f^{-1}(C)=[-2,-1]\cup [1,2]$$

since these are all the values of $x$ where $f(x)$ lands in $[1,4]$.