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# 5.2: Deﬁnition of Functions

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## 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?

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

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.

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

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 $$h(x) = -\sqrt{7-4x+4x^2}$$.

(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 $$\{1,2,3,4,5,6\}$$.

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), $$x=6$$.

(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 $$x=-21$$.