Functions

Section 7.1 Functions

Functions are familar mathematical objects from algebra and calculus. We also saw them in Section 1.3.

Definition 7.1.1.

A function, $f:X\rightarrow Y\text{,}$ is a map in which each input is related to one and only one output.

We say $X$ is the domain and $Y$ is the codomain of $f\text{.}$

Example 7.1.2. Exploring the Definition of a Function.

Let $f: X\rightarrow Y$ be a map as shown in the figure.

Figure 7.1.3 shows a map that is not a function since $x_2$ maps to two different outputs.

Let $g: X\rightarrow Y$ be a map as shown in the figure.

Figure 7.1.4 shows a map that is not a function since $x_1$ does not map to any output.

Let $h: X\rightarrow Y$ be a map as shown in the figure.

Figure 7.1.5 shows a map that is a function since each $x_i$ maps to exactly one $y_i\text{.}$

For a given $x\in X\text{,}$ $f(x)$ is

the output of $f\text{,}$
the value of $f$ at $x\text{,}$
the image of $x$ under $f\text{.}$

Definition 7.1.6.

The image or range of a set $X$ under $f$ is the set of outputs of $f$ corresponding to inputs from $X\text{.}$ In notation

\begin{equation*} \text{Im}(f)=\{y\in Y : y=f(x) \text{ for some } x\in X\}. \end{equation*}

If $f(x)=y$ we say $x$ is a preimage or an inverse image of $y\text{.}$

Since $y$ can have several preimages, we usually care about the set of all of them.

Definition 7.1.7.

Let $f:X\rightarrow Y$ be a function. The set $\{x\in X : f(x)=y\}$ is the preimage of $y\text{.}$

Activity 7.1.1.

Define $f:\mathbb{Z}\rightarrow\mathbb{Z}$ by $f(n)=r$ where $r$ is the remainder when $n$ is divided by 3.

(a)

Find $f(0), f(9), f(5), f(-7), f(10)\text{.}$

(b)

What is the image of $\mathbb{Z}$ under $f\text{?}$

(c)

What is the set of preimages of 0? In other words, find the preimage of 0.

If we have a map from a finite set to a finite set, we can draw an arrow diagram in which we use arrows to represent the map from $X$ to $Y\text{,}$ as in Example 7.1.2.

Example 7.1.8. Arrow Diagram.

Let $X=\{a, b, c, d\}$ and $Y=\{0, 1, 2\}\text{.}$ Let $f: X\rightarrow Y$ be the function given by the following arrow diagram.

Find the domain of $f\text{.}$

Answer 1.

$\{a, b, c, d\}$

Find the codomain of $f\text{.}$

Answer 2.

$\{0, 1, 2\}$

Find the range or image of $f\text{.}$

Answer 3.

$\{0, 2\}$

Find $f(a)\text{.}$

Answer 4.

$0$

Find the preimage or inverse image of $0\text{.}$

Answer 5.

$\{a, b\}$

Example 7.1.10. Finding Sets for a Function.

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be given by $f(x)=x^2\text{.}$

Find the domain of $f\text{.}$

Answer 1.

$\mathbb{R}$

Find the codomain of $f\text{.}$

Answer 2.

$\mathbb{R}$

Find the range of $f\text{.}$

Answer 3.

$[0, \infty)$

Find the preimage (or inverse image) of $1\text{.}$

Answer 4.

$\{1, -1\}$

Let $f, g$ be functions from $X$ to $Y\text{.}$ Then $f=g$ if $f(x)=g(x)$ for all $x\in X\text{.}$

In this course we want to look at functions to and from sets other than just the real numbers. For example, we may have functions from finite sets to finite sets.

Example 7.1.11. More Functions.

A sequence is a function from $\mathbb{Z}^+$ to $\mathbb{R}\text{.}$ For example, $f(n)=\frac{1}{n}\text{.}$

We may also have functions involving Cartesian products of sets. For example, $f:\mathbb{Z}\times\mathbb{Z}\rightarrow\mathbb{Z}$ given by $f(a, b)=a+b\text{.}$

Activity 7.1.2.

Define $f:\mathbb{Z}\times\mathbb{Z}\rightarrow\mathbb{Z}$ by $f(a, b)=a-b\text{.}$

(a)

Find $f(0, 2), f(1, 1), f(2, 0), f(3, 2), f(n, 0)\text{.}$

(b)

What is the image of $\mathbb{Z}\times\mathbb{Z}$ under $f\text{?}$

(c)

What is the preimage of 0?

Activity 7.1.3.

Define $f:\mathbb{Z}\rightarrow\mathbb{Z}\times\mathbb{Z}$ by $f(a)=(a, a)\text{.}$

(a)

Find $f(0), f(2), f(-3)\text{.}$

(b)

What is the image of $\mathbb{Z}$ under $f\text{?}$ Is $(1, 1)$ in the image? Is $(-1, 1)$ in the image?

(c)

What is the preimage of $(0, 0)\text{?}$

Since a function needs to satisfy the property that each $x\in X$ can only map to one $y \in Y\text{,}$ we say a function is well-defined if whenever $a=b\text{,}$ $f(a)=f(b)\text{.}$ Most of the functions you’ve seen in algebra and calculus are clearly well-defined since when $a=b\text{,}$ $f(a)=f(b)\text{.}$ This property is really only interesting when elements of $X$ have multiple representations. In other words, when two equal elements in $X$ have two different forms. The most familiar set where this happens is $\mathbb{Q}\text{.}$ For example, $\frac{1}{2}=\frac{2}{4}\text{.}$

Example 7.1.12. A Map That is Not Well-Defined.

Let $f:\mathbb{Q}\rightarrow \mathbb{Z}$ be given by $f(p/q)=p+q\text{.}$

Then $1/2=2/4$ in $\mathbb{Q}\text{,}$ but $f(1/2)=1+2=3$ and $f(2/4)=2+4=6\text{.}$ Thus, $1/2=2/4$ but $f(1/2)\neq f(2/4)\text{.}$

Thus $f$ is not well-defined, and hence, $f$ is not a function.

Activity 7.1.4.

Let $f:\mathbb{Q}\rightarrow\mathbb{Z}$ be given by $f(m/n)=m\text{.}$ Show $f$ is not well-defined by finding two equivalent fractions in $\mathbb{Q}$ that map to two different integers.