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 \to Y,

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 .

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Example 7.1.2. Exploring the Definition of a Function.

Let

f : X \to 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 \to 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 \to 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} .

For a given

x \in X,

f (x)

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the output of $f,$
the value of $f$ at $x,$
the image of $x$ under $f .$

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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 .

In notation

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Im (f) = {y \in Y : y = f (x) for some x \in X} .

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f (x) = y

we say

x

is a preimage or an inverse image of

y .

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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 \to Y

be a function. The set

{x \in X : f (x) = y}

is the preimage of

y .

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Activity 7.1.1.

Define

f : Z \to Z

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) .

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(b)

What is the image of

Z

under

f ?

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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

Y,

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} .

Let

f : X \to Y

be the function given by the following arrow diagram.

Find the domain of

f .

Answer 1.

{a, b, c, d}

Find the codomain of

f .

Answer 2.

{0, 1, 2}

Find the range or image of

f .

Answer 3.

{0, 2}

Find

f (a) .

Answer 4.

0

Find the preimage or inverse image of

0 .

Answer 5.

{a, b}

Example 7.1.10. Finding Sets for a Function.

Let

f : R \to R

be given by

f (x) = x^{2} .

Find the domain of

f .

Answer 1.

R

Find the codomain of

f .

Answer 2.

R

Find the range of

f .

Answer 3.

[0, \infty)

Find the preimage (or inverse image) of

1 .

Answer 4.

{1, - 1}

Let

f, g

be functions from

X

Y .

Then

f = g

f (x) = g (x)

for all

x \in X .

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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

Z^{+}

R .

For example,

f (n) = \frac{1}{n} .

We may also have functions involving Cartesian products of sets. For example,

f : Z \times Z \to Z

given by

f (a, b) = a + b .

Activity 7.1.2.

Define

f : Z \times Z \to Z

f (a, b) = a - b .

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(a)

Find

f (0, 2), f (1, 1), f (2, 0), f (3, 2), f (n, 0) .

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(b)

What is the image of

Z \times Z

under

f ?

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(c)

What is the preimage of 0?

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Activity 7.1.3.

Define

f : Z \to Z \times Z

f (a) = (a, a) .

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(a)

Find

f (0), f (2), f (- 3) .

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(b)

What is the image of

Z

under

f ?

(1, 1)

in the image? Is

(- 1, 1)

in the image?

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(c)

What is the preimage of

(0, 0) ?

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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,

we say a function is well-defined if whenever

a = b,

f (a) = f (b) .

Most of the functions you’ve seen in algebra and calculus are clearly well-defined since when

a = b,

f (a) = f (b) .

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

Q .

For example,

\frac{1}{2} = \frac{2}{4} .

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

Let

f : Q \to Z

be given by

f (p / q) = p + q .

Then

1 / 2 = 2 / 4

Q,

but

f (1 / 2) = 1 + 2 = 3

and

f (2 / 4) = 2 + 4 = 6 .

Thus,

1 / 2 = 2 / 4

but

f (1 / 2) \neq f (2 / 4) .

Thus

f

is not well-defined, and hence,

f

is not a function.

Activity 7.1.4.

Let

f : Q \to Z

be given by

f (m / n) = m .

Show

f

is not well-defined by finding two equivalent fractions in

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} .

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Let

f : A \to B

be given by

f (1) = 4, f (2) = 8, f (3) = 2 .

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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} .

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Let

f : A \to B

be given by

f (1) = 2, f (2) = 4, f (3) = 2 .

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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} .

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Let

f : B \to C

be given by

f (2) = 0, f (4) = 1 .

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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} .

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Let

f : B \to C

be given by

f (2) = 0, f (4) = 1, f (6) = 1, f (8) = 1 .

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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} .

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Let

f : C \to C

be given by

f (0) = 0, f (0) = 1, f (1) = 0, f (1) = 1 .

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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} .

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Let

f : A \times C \to C

be given by

f (a, c) = c .

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Which of the following are in the preimage of

0 \in C .

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

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Exercises Exercises

1.

Let

X = {1, 3, 5}

and

Y = {a, b, c, d} .

Define the function

g : X \to Y

as the set of ordered pairs

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{(1, b), (3, b), (5, b)} .

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Write the domain and codomain of $g .$
What is the range of $g ?$
Is 3 an inverse image of $a ?$ Is 1 an inverse image of $b ?$
What is the inverse image of $b ?$ What is the inverse image of $c ?$
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.
If two elements in the codomain of a function are equal, then their preimages in the domain are equal.
A function can have the same output for more than one input.
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 : P (A) \to Z

where for all sets

X

P (A),

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

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

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4.

Let

D

be the set of all finite subsets of positive integers. Let

T : Z^{+} \to D

be the function where for each positive integer

n,

T (n)

is the set of positive divisors of

n .

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Find $T (1) .$
Find $T (15) .$
Find $T (17) .$
Find $T (5) .$
Find $T (18) .$
Find $T (21) .$

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5.

Define the function

F : Z \times Z \to Z \times Z

where for all ordered pairs

(a, b)

of integers,

F (a, b) = (2 a + 1, 3 b - 2) .

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Find $F (4, 4) .$
Find $F (2, 1) .$
Find $F (3, 2) .$
Find $F (1, 5) .$

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6.

Let

X = {1, 2, 3, 4}

and

Y = {a, b, c, d, e} .

Define

g : X \to Y

g (1) = a, g (2) = a, g (3) = a

and

g (4) = d .

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

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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 $g : Q \to Z$ by the rule

$g (\frac{m}{n}) = m - n$

for all integers $m$ and $n$ with $n \neq 0 .$
Define $h : Q \to Q$ by the rule

$h (\frac{m}{n}) = \frac{m^{2}}{n}$

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

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