Section 1.3 Functions
Subsection 1.3.1 Basic definitions
In the Pre-Calculus or Calculus courses, functions are introduced as rules that assign a unique output \(f(x)\) (in a set \(B\) called the range) to each input \(x\) in a set \(A\) called the domain. The notation \(f:A\longrightarrow B\) is used, where \(f\) is the name of the function.
But in a more rigorous treatment, we rely exclusively on the notions of set theory, and avoid concepts such as “a rule” that have not been clearly defined. Recall the notion of Cartesian product from the previous section.
This definition is less unfamiliar than it may appear at first sight: think of the graph of a function as you have studied it in Pre-Calculus or Calculus. The graph is the set of all pairs \((x,f(x))\) as \(x\) ranges in the domain of \(f\text{.}\) So the graph of a function is nothing but a certain collection of pairs in the Cartesian product \(A\times B\text{,}\) and the condition in our new definition is just telling us that this graph must satisfy the vertical line test. Definition 1.3.1 is just saying that a function is a graph with a certain condition (the condition that it must satisfy the vertical line test). When we take the new point of view, the notation \(y=f(x)\) is only a convention to say that \((x,y)\in f\text{.}\)
Definition 1.3.2.
Suppose \(f:A\longrightarrow B\text{.}\) An element of \(A\) is called an input and an element \(y\) of \(B\) for which there is some \(x\in A\) such that \(y=f(x)\) is called an output. The subset of the co-domain \(B\) consisting of all the outputs is called the range of \(f\text{,}\) and denoted by \({\cal R}(f)\text{.}\) In symbols, \({\cal R}(f) = \{f(x): x\in A\}\text{.}\)Subsection 1.3.2 Properties of functions
We will now list many definitions summarizing the most important notions and properties of functions that are used in this course. The exercises contain several examples and results based on these definitions.
Definition 1.3.3.
A function \(f:A\longrightarrow B\) is called injective or one-to-one if distinct inputs will produce distinct outputs. In symbols, \(f\) is injective if
\begin{equation*}
(\forall x_1,x_2\in A)(f(x_1)=f(x_2)
\Rightarrow x_1=x_2)\text{.}
\end{equation*}
Definition 1.3.4.
A function \(f:A\longrightarrow B\) is called surjective or onto if every element of the co-domain is in fact an output. In symbols, \(f\) is surjective if
\begin{equation*}
(\forall y\in B)(\exists x\in A)(f(x)=y)\text{.}
\end{equation*}
Note that an equivalent way to say that a function is surjective is to say that the range is equal to the co-domain, or in symbols \(f(X)=Y\text{.}\)
Definition 1.3.5.
Suppose \(f:A\longrightarrow B\text{.}\) If \(X\subset A\text{,}\) the image of \(X\) is the subset \(f(X)\) of \(B\) consisting of all the output values corresponding to the inputs that are in \(X\text{.}\) In symbols,
\begin{equation*}
f(X)=\{f(x):x\in X\}.
\end{equation*}
If \(Y\subset B\text{,}\) the pre-image of \(Y\) is the subset \(f^{-1}(Y)\) of \(A\) consisting of all the input values whose outputs are in \(Y\text{.}\) In symbols,
\begin{equation*}
f^{-1}(Y)=\{x\in A: f(x)\in Y\}.
\end{equation*}
Example 1.3.6.
Suppose \(f: {\mathbb Q}\longrightarrow {\mathbb Q}\) is defined by \(f(x)=x^2\) (or, according to Definition 1.3.1, \(f=\{(x,x^2):x\in {\mathbb Q}\}\text{.}\) Then \(f\) is neither injective nor surjective.
Example 1.3.7.
Let now \({\mathbb Q}^+\) be the set of non-negative rational numbers, \({\mathbb Q}^+=\{x\in {\mathbb Q}:x\geq 0\}\text{,}\) and \(f: {\mathbb Q}^+ \longrightarrow {\mathbb Q}\) , \(f(x)=x^2\text{.}\) Then \(f\) is injective, but it is not surjective.
Example 1.3.8.
Let \(f: {\mathbb Q} \longrightarrow {\mathbb Q}^+\) , \(f(x)=x^2\text{.}\) Then \(f\) is surjective, but it is not injective.
Example 1.3.9.
Let \(f: {\mathbb Q}^+ \longrightarrow {\mathbb Q}^+\) , \(f(x)=x^2\text{.}\) Then \(f\) is both injective and surjective.
Definition 1.3.10.
Let \(A\) be a set. The identity function on \(A\) is the function \(\mbox{id}_A: A\longrightarrow A\) defined by \(\mbox{id}_A(x)=x\) for all \(x\in A\text{.}\) So in terms of subsets, \(\mbox{id}_A=\{(x,x): x\in A\}\) .Definition 1.3.11.
Let \(A,B,C\) be sets. Suppose \(f:A\longrightarrow B\) and \(g:B\longrightarrow C\) are functions. The composition of \(f\) and \(g\) is the function \(f\circ g: A\longrightarrow C\) defined by
\begin{equation*}
(f\circ g) (x) =f(g(x)).
\end{equation*}
Definition 1.3.12.
Let \(A,B\) be sets, and \(f:A\longrightarrow B\text{.}\) We say that a function \(g:B\longrightarrow A\) is a right inverse of \(f\) if \(f\circ g=\mbox{id}_B\text{,}\) and we say that \(g\) is a left inverse of \(f\) if \(g\circ f=\mbox{id}_A\text{.}\)So \(g:B\longrightarrow A\) is a right inverse of \(f: A\longrightarrow B\) means that \(f(g(y))=y\) for all \(y\in B\text{,}\) and \(g\) is a left inverse of \(f\) if \(g(f(x))=x\) for all \(x\in A\text{.}\)
