5.2: Definition of Functions

Definition: Function

Let \(A\) and \(B\) be nonempty sets. A function from \(A\) to \(B\) is a rule that assigns to every element of \(A\) a unique element in \(B\). We call \(A\) the domain, and \(B\) the codomain, of the function. If the function is called \(f\), we write \(f :A \to B\). Given \(x\in A\), its associated element in \(B\) is called its image under \(f\). In other words, a function is a relation from \(A\) to \(B\) with the condition that for every element in the domain, there exists a unique image in the codomain (this is really two conditions: existence of an image and uniqueness of an image). We denote it \(f(x)\), which is pronounced as “ \(f\) of \(x\).”

A function is sometimes called a map or mapping. Hence, we sometimes say \(f\) maps \(x\) to its image \(f(x)\).

Example \(\PageIndex{1}\label{eg:defnfcn-01}\)

The function \(f:\{a,b,c\} \) to \(\{1,3,5,9\}\) is defined according to the rule \[f(a)=1, \qquad f(b)=5, \qquad\mbox{and}\qquad f(c) = 9.\] It is a well-defined function. The rule of assignment can be summarized in a table: \[\begin{array}{|c||c|c|c|} \hline x & a & b & c \\ \hline f(x)& 1 & 5 & 9 \\ \hline \end{array}\] We can also describe the assignment rule pictorially with an arrow diagram, as shown in Figure 6.2.

The two key requirements of a function are

• every element in the domain has an image under \(f\), and
• the image is unique.

You may want to remember that every element in \(A\) has exactly one “partner” in \(B\).

Example \(\PageIndex{2}\label{eg:defnfcn-02}\)

Figure 6.3 depicts two examples of non-functions. In the one on the left, one of the elements in the domain has no image associated with it; thus lacking existence of an image. In the one on the right, one of the elements in the domain has two images assigned to it; thus lacking uniqueness of an image. Both are not functions.

hands-on exercise \(\PageIndex{1}\label{he:defnfcn-01}\)

Do these rules \[\begin{array}{|c||c|c|c|} \hline x & a & b & c \\ \hline f(x)& 5 & 3 & 3 \\ \hline \end{array} \hskip0.75in \begin{array}{|c||c|c|} \hline x & b & c \\ \hline g(x)& 9 & 5 \\ \hline \end{array} \hskip0.75in \begin{array}{|c||c|c|c|c|} \hline x & a & b & b & c \\ \hline h(x)& 1 & 5 & 3 & 9 \\ \hline \end{array}\] produce functions from \(\{a,b,c\}\) to \(\{1,3,5,9\}\)? Explain.

hands-on exercise \(\PageIndex{2}\label{he:defnfcn-02}\)

Does the definition \[r(x) = \cases{ x & if today is Monday, \cr 2x & if today is not Monday \cr}\] produce a function from \(\mathbb{R}\) to \(\mathbb{R}\)? Explain.

hands-on exercise \(\PageIndex{3}\label{he:defnfcn-03}\)

Does the definition \[s(x) = \cases{ 5 & if $x<2$, \cr 7 & if $x>3$, \cr}\] produce a function from \(\mathbb{R}\) to \(\mathbb{R}\)? Explain.

Example \(\PageIndex{3}\label{eg:defnfcn-03}\)

The function \(f:{[0,\infty)}\to{\mathbb{R}}\) is defined by \[f(x) = \sqrt{x}.\]  Also the function \({g}:{[2,\infty)}\to{\mathbb{R}}\) is defined as \[g(x) = \sqrt{x-2}.\] Can you explain why the domain of \(g\) is \([2,\infty)\)?

Example \(\PageIndex{4}\label{eg:defnfcn-04}\)

Let \(A\) denote the set of students taking Discrete Mathematics, and \(G=\{A,B,C,D,F\}\), and \(\ell(x)\) is the final grade of student \(x\) in Discrete Mathematics. Every student should receive a final grade, and the instructor has to report one and only one final grade for each student. \(\ell:A \to G.\)  This is precisely what we call a function.

Example \(\PageIndex{5}\label{eg:defnfcn-05}\)

The function \({n}:{\mathscr{P}(\{a,b,c,d\})}\to{\mathbb{Z}}\) is defined as \(n(S)=|S|\). It evaluates the cardinality of a subset of \(\{a,b,c,d\}\). For example, \[n\big(\{a,c\}\big) = n\big(\{b,d\}\big) = 2.\] Note that \(n(\emptyset)=0\).

hands-on exercise \(\PageIndex{4}\label{he:defnfcn-04}\)

Consider Example 5.2.5.  What other subsets \(S\) of \(\{a,b,c,d\}\) also yield \(n(S)=2\)? What are the smallest and the largest images the function \(n\) can produce?

Example \(\PageIndex{6}\label{eg:defnfcn-06}\)

Consider a function \({f}:{\mathbb{Z}_7}\to{\mathbb{Z}_5}\). The domain and the codomain are,

\[\mathbb{Z}_7 = \{0,1,2,3,4,5,6\}, \qquad\mbox{and}\qquad \mathbb{Z}_5 = \{0,1,2,3,4\},\]

respectively. Not only are their elements different, their binary operations are different too. In the domain \(\mathbb{Z}_7\), the arithmetic is performed modulo 7, but the arithmetic in the codomain \(\mathbb{Z}_5\) is done modulo 5. So we need to be careful in describing the rule of assignment if a computation is involved. We could say, for example,

\[f(x) = z, \quad\mbox{where } z \equiv 3x \pmod{5}.\]

Consequently, starting with any element \(x\) in \(\mathbb{Z}_7\), we consider \(x\) as an ordinary integer, multiply by 3, and reduce the answer modulo 5 to obtain the image \(f(x)\). For brevity, we shall write

\[f(x) \equiv 3x \pmod{5}.\]

We summarize the images in the following table:

\[\begin{array}{|c||*{7}{c|}} \hline n & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline f(n) & 0 & 3 & 1 & 4 & 2 & 0 & 3 \\ \hline \end{array}\]

Take note that the images start repeating after \(f(4)=2\).

hands-on exercise \(\PageIndex{5}\label{he:defnfcn-05}\)

Tabulate the images of \({g}:{\mathbb{Z}_{10}}\to{\mathbb{Z}_5}\) defined by \[g(x) \equiv 3x \pmod{5}.\]

Definition: A Function as a Set of Ordered Pairs

A function \({f}:{A}\to{B}\) can be written as a set of ordered pairs \((x,y)\) from \(A\times B\) such that \(y=f(x)\).

 A function is, by definition, a set of ordered pairs, with certain restrictions.

Example \(\PageIndex{7}\label{eg:defnfcn-07}\)

The function \(f\) in Example 5.26 can be written as the set of ordered pairs \[\{(0,0), (1,3), (2,1), (3,4), (4,2), (5,0), (6,3)\}.\] If one insists, we could display the graph of a function using an \(xy\)-plane that resembles the usual Cartesian plane. Keep in mind: the elements \(x\) and \(y\) come from \(A\) and \(B\), respectively. We can “plot” the graph for \(f\) in Example 5.26 as shown below.

Besides using a graphical representation, we can also use a \((0,1)\)-matrix. A \((0,1)\)-matrix is a matrix whose entries are 0 and 1. For the function \(f\), we use a \(7\times5\) matrix, whose rows and columns correspond to the elements of \(A\) and \(B\), respectively, and put one in the \((i,j)\)th entry if \(j=f(i)\), and zero otherwise. The resulting matrix is

\[\begin{array}{cc} & \begin{array}{ccccc} 0 & 1 & 2 & 3 & 4 \end{array} \\ \begin{array}{c} 0 \\ 1 \\ 2 \\ 3 \\ 4 \\ 5 \\ 6 \end{array} & \left(\begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}\right) \end{array}\]

We call it the incidence matrix for the function \(f\).

hands-on exercise \(\PageIndex{6}\label{he:defnfcn-06}\)

"Plot” the graph of \(g\) in Hands-On Exercise 5.2.5

Summary and Review

• A function \(f\) from a set \(A\) to a set \(B\) (called the domain and the codomain, respectively) is a rule that describes how a value in the codomain \(B\) is assigned to an element from the domain \(A\).
• But it is not just any rule; rather, the rule must assign to every element \(x\) in the domain a unique value in the codomain.
• This unique value is called the image of \(x\) under the function \(f\), and is denoted \(f(x)\).
• We use the notation \({f}:{A}\to{B}\) to indicate that the name of the function is \(f\), the domain is \(A\), and the codomain is \(B\).
• A function \({f}:{A}\to{B}\) is the collection of all ordered pairs \((x,y)\) from \(A\times B\) such that \(y=f(x)\).
• The graph of a function may not be a curve, as in the case of a real function. It can be just a collection of points.
• We can also display the images of a function in a table, or represent the function with an incidence matrix.

Exercises 

exercise \(\PageIndex{1}\label{ex:defnfcn-01}\)

What subset \(A\) of \(\mathbb{R}\) would you use to make \({f}:{A}\to{\mathbb{R}}\) defined by \(f(x) = \sqrt{3x-7}\) a function?

Answer

\(\big[\frac{7}{3},\infty\big)\)

exercise \(\PageIndex{2}\label{ex:defnfcn-02}\)

What subset \(A\) of \(\mathbb{R}\) would you use to make

1. \({g}:{A}\to{\mathbb{R}}\), where \(g(x) = \sqrt{(x-3)(x-7)}\)
2. \({h}:{A}\to{\mathbb{R}}\), where \(h(x) = \frac{x+2}{\sqrt{(x-2)(5-x)}}\)

 functions?

exercise \(\PageIndex{3}\label{ex:defnfcn-03}\)

Which of these data support a function from \(\{1,2,3,4\}\) to \(\{1,2,3,4\}\)? Explain.

\[\begin{array}{|c||c|c|c|} \hline x & 1 & 2 & 3 \\ \hline f(x) & 3 & 4 & 2 \\ \hline \end{array} \hskip0.4in \begin{array}{|c||c|c|c|c|} \hline x & 1 & 2 & 3 & 4 \\ \hline g(x) & 2 & 4 & 3 & 2 \\ \hline \end{array} \hskip0.4in \begin{array}{|c||c|c|c|c|c|} \hline x & 1 & 2 & 3 & 3 & 4 \\ \hline h(x) & 2 & 4 & 3 & 2 & 3 \\ \hline \end{array}\]
 

Answer

Only \(g\) is a function. The image \(f(4)\) is undefined, and there are two values for \(h(3)\). Hence, both \(f\) and \(h\) are not well-defined functions.

exercise \(\PageIndex{4}\label{ex:defnfcn-04}\)

(a) Use arrow diagrams to show three different functions from \(\{1,2,3,4\}\) to \(\{1,2,3,4\}\).
(b) How many different functions from \(\{1,2,3,4\}\) to \(\{1,2,3,4\}\) are possible? 

exercise \(\PageIndex{5}\label{ex:defnfcn-05}\)

Determine whether these are functions. Explain.

1. \({f}:{\mathbb{R}}\to{\mathbb{R}}\), where \(f(x) = \frac{3}{x^2+5}\).
2. \({g}:{(5,\infty)}\to{\mathbb{R}}\), where \(g(x) = \frac{7}{\sqrt{x-4}}\).
3. \({h}:{\mathbb{R}}\to{\mathbb{R}}\), where \(h(x) = -\sqrt{7-4x+4x^2}\).

Answer

(a) Yes, because no division by zero will ever occur.

exercise \(\PageIndex{6}\label{ex:defnfcn-06}\)

Determine whether these are functions. Explain.

1. \({s}:{\mathbb{R}}\to{\mathbb{R}}\), where \( x^2+[s(x)]^2=9\).
2. \({t}:{\mathbb{R}}\to{\mathbb{R}}\), where \(|x-t(x)|=4\).

exercise \(\PageIndex{7}\label{ex:defnfcn-07}\)

Use arrow diagrams to show two different functions from \(\{a,b,c,d\}\) to \(\{1,2,3,4,5,6\}\).

 

Answer

answers will vary

exercise \(\PageIndex{8}\label{ex:defnfcn-08}\)

Let \(T\) be your family tree that includes your biological mother, your maternal grandmother, your maternal great-grandmother, and so on, and all of their female descendants. Determine which of the following define a function from \(T\) to \(T\).

1. \({h_1}:{T}\to{T}\), where \(h_1(x)\) is the mother of \(x\).
2. \({h_2}:{T}\to{T}\), where \(h_2(x)\) is \(x\)’s sister.
3. \({h_3}:{T}\to{T}\), where \(h_3(x)\) is an aunt of \(x\).
4. \({h_4}:{T}\to{T}\), where \(h_4(x)\) is the eldest daughter of \(x\)’s maternal grandmother.

exercise \(\PageIndex{9}\label{ex:defnfcn-09}\)

For each of the following functions, determine the image of the given \(x\).

1. \({k_1}:{\mathbb{N}-\{1\}}\to{\mathbb{N}}\), \(k_1(x)=\mbox{smallest prime factor of }x \), \(x=217\).
2. \({k_2}:{\mathbb{Z}_{11}}\to{\mathbb{Z}_{11}}\), \(k_2(x)\equiv3x\) (mod 11), \(x=6\).
3. \({k_3}:{\mathbb{Z}_{15}}\to{\mathbb{Z}_{15}}\), \(k_3(x)\equiv3x\) (mod 15), \(x=6\).

Answer

(a) 7 (b) 7 (c) 3

exercise \(\PageIndex{10}\label{ex:defnfcn-10}\)

For each of the following functions, determine the images of the given \(x\)-values.

• \({\ell_1}:{\mathbb{Z}}\to{\mathbb{Z}}\), \(\ell_1(x)=x\bmod7\), \(x=250\), \(x=0\), and \(x=-16\).

Remark: Recall that, without parentheses, the notation “mod” means the binary operation mod.

\({\ell_2}:{\mathbb{Z}}\to{\mathbb{Z}}\), \(\ell_2(x)=\gcd(x,24)\), \(x=100\), \(x=0\), and \(x=-21\).