149 Preimages and Inverse Functions
A preimage is the set of domain elements sent into a given subset that is used together with inverse functions in the study of continuity.
Definition (Inverse Function) A function from the codomain of a function to its domain,
\[ f^{-1} : Y \rightarrow X, \]
where the following condition applies:
- \(f^{-1}(y) = x \iff f(x) = y\).
where
- \(f : X \rightarrow Y\) is a one-to-one and onto function.
- \(X\) is the domain of \(f\).
- \(Y\) is the codomain of \(f\).
- \(x \in X, y \in Y\).
Definition (Preimage) A subset of the domain of a function,
\[ f^{-1}(Y_{subset}) \subseteq X, \]
where the following condition applies:
- \(f^{-1}(Y_{subset}) = \{x \in X : f(x) \in Y_{subset}\}\).
where
- \(f : X \rightarrow Y\) is a function.
- \(X\) is the domain of \(f\).
- \(Y\) is the codomain of \(f\).
- \(Y_{subset} \subseteq Y\).
Definition (Continuity) A property of a map between open subsets, \(f: X \rightarrow Y\), where the following conditions apply:
- The preimage of every open subset of \(Y\) is an open subset of \(X\).
where
- \(X, Y \subseteq \mathbb{R}^n\)
- \(f^{-1}(U) = \{x \in X : f(x) \in U\}\)
- \(f^{-1} : \mathcal{P}(Y) \rightarrow \mathcal{P}(X)\) is the preimage map from subsets of \(Y\) to subsets of \(X\).
- \(U \subseteq Y\)
- \(\mathcal{P}(X)\) is the power set of \(X\)
Definition (Surjective) A property of a function from a set \(X\) to a set \(Y\), \(F: X \rightarrow Y\), where the following condition applies:
- For every element \(y\) in \(Y\), there exists an element \(x\) in \(X\) such that \(F(x) = y\).
where
- \(X\) is the domain of \(F\)
- \(Y\) is the co-domain of \(F\)
- \(F(X) = Y\) is the statement that the image equals the co-domain.
- \(F(X) = \{y \in Y : y = F(x) \text{ for some } x \in X\}\) is the image of \(X\) under \(F\).
Note:
- \(F(X) = Y\) means the range of the function equals its co-domain.
Definition (Injective) A property of a function from a set \(X\) to a set \(Y\), \(F: X \rightarrow Y\), where the following condition applies:
- For all elements \(x_1\) and \(x_2\) in \(X\), if \(F(x_1) = F(x_2)\), then \(x_1 = x_2\).
where
- \(X\) is the domain of \(F\).
- \(Y\) is the co-domain of \(F\).
- \(x_1, x_2\) are elements of \(X\).
Note:
- The condition is logically equivalent to saying that if \(x_1 \neq x_2\), then \(F(x_1) \neq F(x_2)\).
- For an injective function, each element of the co-domain is the image of at most one element of the domain.
Definition (Bijective) A property of a function from a set \(X\) to a set \(Y\), \(F: X \rightarrow Y\), where the following conditions apply:
- The function \(F\) is injective
- The function \(F\) is surjective
where
- \(X\) is the domain of \(F\).
- \(Y\) is the co-domain of \(F\)
149.1 Elementary Example
149.1.1 Simple
A preimage collects inputs; an inverse function exists only for a bijection and returns single inputs.
\[ f : X \rightarrow Y \]
\[ X = \{ 1,\ 2,\ 3 \},\quad Y = \{ a,\ b,\ c \} \]
\[ f(1)=a,\ f(2)=b,\ f(3)=c \]
\[ f^{-1}(\{a,b\}) = \{ 1,\ 2 \},\quad f^{-1}(a) = 1 \]
where
- \(f^{-1}(\{a,b\})\) is a preimage set.
- \(f^{-1}(a)\) is the inverse-function value at \(a\).
149.1.2 General
If \(f\) is not injective, preimages of points may have several elements, so no inverse function exists.
\[ X = \{ 1,\ 2,\ 3,\ 4 \},\quad Y = \{ a,\ b \} \]
\[ f(1)=a,\ f(2)=a,\ f(3)=b,\ f(4)=b \]
\[ f^{-1}(\{a\}) = \{ 1,\ 2 \} \]
where
- \(f^{-1}(\{a\})\) is a preimage with two elements.
- \(f\) is not bijective, so \(f^{-1}\) as a function \(Y \rightarrow X\) does not exist.