Preimage
High school
Consider a function $f$ and a set $B$ of possible outputs. The preimage of $B$ under $f$ is the set of all inputs $x$ whose output $f(x)$ is in $B$.
Undergraduate
Consider a function $f : X \to Y$ and a set $B \subseteq Y$. The preimage of $B$ is
\[f^{-1}(B) = \setdef{x \in X \suchthat f(x) \in B}.\]Examples
Worked examples
-
$f(x) = x^2$ on $\reals$
: $f^{-1}(\setdef{4}) = \setdef{-2, 2}$ and $f^{-1}(\setdef{-1}) = \emptyset$
High school
The preimage $f^{-1}(\setdef{4})$ exists even though $f$ has no inverse. No real number squares to $-1$, so $f^{-1}(\setdef{-1})$ is empty.
Notes
The preimage is always defined, even for functions with no inverse.