Here is a summary of all the main concepts and definitions we use when working with functions.
A function is a rule that assigns each element of a set, called the domain, to exactly one element of a second set, called the codomain.
Notation: \(f:X \to Y\) is our way of saying that the function is called \(f\text{,}\) the domain is the set \(X\text{,}\) and the codomain is the set \(Y\text{.}\)
To specify the rule for a function with small domain, use two-line notation by writing a matrix with each output directly below its corresponding input, as in:
\(f(x) = y\) means the element \(x\) of the domain (input) is assigned to the element \(y\) of the codomain. We say \(y\) is an output. Alternatively, we call \(y\) the image of \(x\) under \(f\).
The range is a subset of the codomain. It is the set of all elements which are assigned to at least one element of the domain by the function. That is, the range is the set of all outputs.
A function is injective (an injection or one-to-one) if every element of the codomain is the image of at most one element from the domain.
A function is surjective (a surjection or onto) if every element of the codomain is the image of at least one element from the domain.
A bijection is a function which is both an injection and surjection. In other words, if every element of the codomain is the image of exactly one element from the domain.
The image of an element \(x\) in the domain is the element \(y\) in the codomain that \(x\) is mapped to. That is, the image of \(x\) under \(f\) is \(f(x)\text{.}\)
The complete inverse image of an element \(y\) in the codomain, written \(f\inv(y)\text{,}\) is the set of all elements in the domain which are assigned to \(y\) by the function.
The image of a subset \(A\) of the domain is the set \(f(A) = \{f(a) \in Y \st a \in A\}\text{.}\)
The inverse image of a subset \(B\) of the codomain is the set \(f\inv(B) = \{x \in X \st f(x) \in B\}\text{.}\)