Functions

Section 7.1 Functions

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

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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.

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

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Example 7.1.2. Exploring the Definition of a Function.

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

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

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

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

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

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

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

the output of $f\text{,}$

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the value of $f$ at $x\text{,}$

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the image of $x$ under $f\text{.}$

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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*}

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

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

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Definition 7.1.7.

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

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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.

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(a)

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

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(b)

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

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(c)

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

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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.

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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.

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

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Answer 1.

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

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

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Answer 2.

$\{0, 1, 2\}$

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

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Answer 3.

$\{0, 2\}$

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

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Answer 4.

$0$

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

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Answer 5.

$\{a, b\}$

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Example 7.1.10. Finding Sets for a Function.

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

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

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Answer 1.

$\mathbb{R}$

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

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Answer 2.

$\mathbb{R}$

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

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Answer 3.

$[0, \infty)$

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

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Answer 4.

$\{1, -1\}$

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

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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.

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Example 7.1.11. More Functions.

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

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

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Activity 7.1.2.

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

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(a)

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

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(b)

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

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(c)

What is the preimage of 0?

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Activity 7.1.3.

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

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(a)

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

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(b)

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

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(c)

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

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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, $\frac{1}{2}=\frac{2}{4}\text{.}$

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

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

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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 $f(1/2)\neq f(2/4)\text{.}$

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

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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.

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Reading Questions Check Your Understanding

1.

Let $A=\{1, 2, 3\}, B=\{2, 4, 6, 8\}\text{.}$

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Let $f:A\rightarrow B$ be given by $f(1)=4, f(2)=8, f(3)=2\text{.}$

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True or false: $f$ is a function.

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True.
False.

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2.

Let $A=\{1, 2, 3\}, B=\{2, 4, 6, 8\}\text{.}$

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Let $f:A\rightarrow B$ be given by $f(1)=2, f(2)=4, f(3)=2\text{.}$

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True or false: $f$ is a function.

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True.
False.

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3.

Let $B=\{2, 4, 6, 8\}, C=\{0, 1\}\text{.}$

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Let $f:B\rightarrow C$ be given by $f(2)=0, f(4)=1\text{.}$

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True or false: $f$ is a function.

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True.
$6, 8$ don’t map anywhere.
False.
$6, 8$ don’t map anywhere.

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4.

Let $B=\{2, 4, 6, 8\}, C=\{0, 1\}\text{.}$

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Let $f:B\rightarrow C$ be given by $f(2)=0, f(4)=1, f(6)=1, f(8)=1\text{.}$

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True or false: $f$ is a function.

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True.
False.

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5.

Let $C=\{0, 1\}\text{.}$

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Let $f:C\rightarrow C$ be given by $f(0)=0, f(0)=1, f(1)=0, f(1)=1\text{.}$

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True or false: $f$ is a function.

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True.
0 maps to two different values.
False.
0 maps to two different values.

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6.

Let $A=\{1, 2, 3\}, C=\{0, 1\}\text{.}$

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Let $f:A\times C\rightarrow C$ be given by $f(a, c)=c\text{.}$

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Which of the following are in the preimage of $0\in C\text{.}$

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(1, 0)
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$(1, 0)$ is in the preimage.
(2, 0)
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$(2, 0)$ is in the preimage.
(3, 0)
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$(3, 0)$ is in the preimage.
(0, 0)
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$(0, 0)$ is not in $A\times C\text{.}$
0
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$0$ is not in $A\times C\text{.}$
(0, 1)
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$(0, 1)$ is not in $A\times C\text{,}$ and would not map to 0.
1
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$1$ is not in $A\times C\text{.}$

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Exercises Exercises

1.

Let $X=\{1, 3, 5\}$ and $Y=\{a, b, c, d\}\text{.}$ Define the function $g:X\rightarrow Y$ as the set of ordered pairs

\begin{equation*} \{(1, b), (3, b), (5, b)\}. \end{equation*}

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Write the domain and codomain of $g\text{.}$
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What is the range of $g\text{?}$
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Is 3 an inverse image of $a\text{?}$ Is 1 an inverse image of $b\text{?}$
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What is the inverse image of $b\text{?}$ What is the inverse image of $c\text{?}$
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Represent $g$ as an arrow diagram.
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2.

Determine if the following are true or false. Justify your answer.

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If two elements in the domain of a function are equal, then their images in the codomain are equal.
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If two elements in the codomain of a function are equal, then their preimages in the domain are equal.
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A function can have the same output for more than one input.
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A function can have the same input for more than one output.
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3.

Let $A=\{1, 2, 3, 4, 5\}$ and define a function $F:{\cal P}(A)\rightarrow \mathbb{Z}$ where for all sets $X$ in ${\cal P}(A)\text{,}$

\begin{equation*} F(X)= \begin{cases} 0 &\text{if $X$ has an even number of elements}\\ 1 &\text{if $X$ has an odd number of elements.} \end{cases} \end{equation*}

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Find $F(\{1, 3, 4\})\text{.}$
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Find $F(\emptyset)\text{.}$
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Find $F(\{2, 3\})\text{.}$
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$F(\{2, 3, 4, 5\})\text{.}$
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4.

Let $D$ be the set of all finite subsets of positive integers. Let $T:\mathbb{Z}^+\rightarrow D$ be the function where for each positive integer $n\text{,}$ $T(n)$ is the set of positive divisors of $n\text{.}$

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Find $T(1)\text{.}$
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Find $T(15)\text{.}$
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Find $T(17)\text{.}$
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Find $T(5)\text{.}$
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Find $T(18)\text{.}$
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Find $T(21)\text{.}$
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5.

Define the function $F:\mathbb{Z}\times\mathbb{Z}\rightarrow \mathbb{Z}\times\mathbb{Z}$ where for all ordered pairs $(a, b)$ of integers, $F(a, b)=(2a+1, 3b-2)\text{.}$

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Find $F(4, 4)\text{.}$
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Find $F(2, 1)\text{.}$
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Find $F(3, 2)\text{.}$
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Find $F(1, 5)\text{.}$
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6.

Let $X=\{1, 2, 3, 4\}$ and $Y=\{a, b, c, d, e\}\text{.}$ Define $g:X\rightarrow Y$ by $g(1)=a, g(2)=a, g(3)=a$ and $g(4)=d\text{.}$

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Draw an arrow diagram for $g\text{.}$
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Let $A=\{2, 3\}, C=\{a\}\text{,}$ and $D=\{b, c\}\text{.}$ Find $g(A), g(X), g^{-1}(C), g^{-1}(D)\text{,}$ and $g^{-1}(Y)\text{.}$
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7.

Show that each of the following maps is not a function by showing it is not well-defined.

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Define $g:\mathbb{Q}\rightarrow\mathbb{Z}$ by the rule

\begin{equation*} g\Big(\frac{m}{n}\Big)=m-n \end{equation*}

for all integers $m$ and $n$ with $n\neq 0\text{.}$

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Define $h:\mathbb{Q}\rightarrow\mathbb{Q}$ by the rule

\begin{equation*} h\Big(\frac{m}{n}\Big)=\frac{m^2}{n} \end{equation*}

for all integers $m$ and $n$ with $n\neq 0\text{.}$

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