What is a Function?

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one additional rule

Functions

Functions are special relations. While a relation is just two sets of items and some pairings between the two sets, functions satisfy one rule.

THE Rule

Functions are those relations where each domain item is paired with exactly one codomain item, or
Functions are those relations where each domain item occurs in exactly one pair.

Function

A function is a package of three sets

domain
codomain
pairings

such that each domain item is included in one and only one pair.

Mathematicians have a funny way of saying yes or no about a relation possibly being a function. If a supposed function satisfies our one and only rule, then it is said to be a well-defined function. Otherwise, it is not well-defined (meaning, it is not a function).

Well-Defined

A relation is a function and is said to be well-defined, if it satisfies the one and only rule for a function.

Video: Introduction to Functions

[ Click on the arrow to the right to expand for the video. ]

SSN
The SSN function has a domain consisting of all U.S. citizens and a codomain of all 9-digit numbers. SSN pairs a U.S. citizen in the domain with the 9-digit number in the codomain that was issued to them as their social security number.

This function is not well-defined, because there are U.S. citizens without a social security number, like many new-born babies.

ReverseSSN
The ReverseSSN function has a domain consisting of all 9-digit numbers and a codomain consisting of all U.S. citizens. ReverseSSN pairs a number in the domain with the person in the codomain who was issued the number as their social security number.

This function is not well-defined, because there are 9-digit numbers that have not been issued as social security numbers.

RationalAdd
The RationalAdd function has a domain consisting of all rational numbers and a codomain consisting of all integers. RationalAdd pairs a rational number in the domain with an integer in the codomain according to the following procedure:

Since a chosen domain number is a rational number, it can be represented as $\tfrac {A}{B}$ , where $A$ and $B$ are integers. This domain rational number is paired with $A+B$ in the codomain.

This function is not well-defined.

For instance: There is a rational number that can be represented as $\tfrac {2}{3}$ and thus it would be paired with $5$ . However, this same rational number can also be represented as $\tfrac {4}{6}$ and thus would be paired with $10$ . This violates the only rule we have.

This particular rational number from the domain is paired with both $5$ and $10$ . That isn’t allowed for a function. Each domain item must be paired with exactly one codomain number. RationalAdd is not well-defined and not a function.

When function values depend on how you write numbers, that is a pretty good clue that the function may not be well-defined.

SuperBowlWinner
The SuperBowlWinner function has a domain consisting of the NFL Superbowls played and a codomain consisting of all of the NFL teams that have existed. SuperBowlWinner pairs a Superbowl in the domain with the NFL team in the codomain that won that Superbowl.

This function is well-defined. Each Superbowl has exactly one winner.

Whew!

We finally found a well-defined function. That one little rule actually narrows the pool of relations quite a bit. By focusing our investigation on only functions, this rule will help us study relationships between all kinds of measurements (our main goal).

Function Notation

Now that we are only investigating functions, rather than all relations, we discover some opportunities to help our communication. For instance, it turns out that when you select a domain item in a function, you have automatically selected a codomain item.

Our Rule: Each domain item is paired with EXACTLY one codomain item. Not 0. Not 2. Not 3. Exactly 1!

Thus, when you pick a domain item, you have automatically selected its partner in the codomain. We could certainly hunt down this partner inside the codomain. But, we can also just talk about

“the domain member’s partner in the codomain.”

We have notation for this thought.

Function Notation

Let functionName be the name of a function.
Let d represent a domain item of functionName.

Then, “functionName(d)” represents d’s partner in the codomain.

$(d, functionName(d))$ is a pair in functionName.

$functionName(d))$ is called “the VALUE of functionName AT d”.
$functionName(d))$ is called “the IMAGE of d under functionName”.

Language:
$functionName(d))$ is pronounced as “functionName OF d”.

SuperBowlWinner

SuperBowlWinner(Superbowl 13) represents the winning team from Superbowl 13, which is the Pittsburgh Steelers. The following equality is communicating this with an equation.

SuperBowlWinner(Superbowl 13) = Pittsburgh Steelers

The pair (Superbowl 13, SuperBowlWinner(Superbowl 13)) is the pair (Superbowl 13, Pittsburgh Steelers).

Note: The one and only rule for functions doesn’t refer to the codomain.

Not every NFL team has won a Superbowl. As of 2023, the Cleveland Browns had not won a Superbowl. So, there are codomain items that are not partnered with a domain item. This is not true of the domain. Every domain item is paired with a codomain item. That is our one and only function rule. Every domain item appears in exactly one pair. But not every codomain item must appear in a pair. This is significant and deserves some language.

Range

The range of a function is the collection of items in the codomain which are paired with some item in the domain. The range is the collection of function values.

The range and codomain are separate sets. The range and codomain could be equal sets or the range could be a proper subset of the codomain.

$Range \subseteq Codomain$

The Cleveland Browns are not in the range of SuperBowlWinner.
The Pittsburgh Steelers are in the range of SuperBowlWinner.

$\subseteq$ is the symbol for is a subset of.
$\subset$ is the symbol for is a proper subset of.
$\in$ is the symbol for is a member of.
$\mathbb {R}$ is the symbol for the real numbers.
$\mathbb {Q}$ is the symbol for the rational numbers.
$\mathbb {Z}$ is the symbol for the integers.
$\mathbb {N}$ is the symbol for the natural numbers.

For example, $\pi \in \mathbb {R}$ , $\tfrac {2}{3} \in \mathbb {Q}$ , $-3 \in \mathbb {Z}$ , $5 \in \mathbb {N}$ .

For Example, $\mathbb {N} \subseteq \mathbb {Z} \subseteq \mathbb {Q} \subseteq \mathbb {R}$

For Example, $\mathbb {N} \subset \mathbb {Z} \subset \mathbb {Q} \subset \mathbb {R}$

Let $D = \{ 3, 5, 7, 8, 10, 11, 13, 15, 16 \}$
Let $T = \{ 3, 10, 11, 16 \}$
Let $R = \{ 3, 6, 10, 11, 16 \}$
Let $W = \{ 3, 8, 10 \}$

$3 \in D$ $3 \in T$ $3 \in R$ $3 \in W$ $T \subset D$ $T \subseteq D$ $D \subseteq D$ $W \nsubseteq R$ $8 \notin R$

Starring

Let’s name the following function Starring.

domain = { Casablanca, House of Wax, The Godfather, Lawrence of Arabia, Toy Story, The Fly }

codomain = { Marlon Brando, Vincent Price, Humphrey Bogart, Peter O’Toole, Harrison Ford, Al Pacino }

pairs = { (The Godfather, Marlon Brando), (House of Wax, Vincent Price), (Casablanca, Humphrey Bogart), (Lawrence of Arabia, Peter O’Toole), (The Fly, Vincent Price) }

Is Starring well-defined?

Yes No

Remember that every domain element must be used in exactly one ordered pair. Toy Story is in the domain and not in any pair.

StarringAgain

Let’s name the following function StarringAgain.

domain = { Casablanca, House of Wax, The Godfather, Lawrence of Arabia, The Fly }

codomain = { Marlon Brando, Vincent Price, Humphrey Bogart, Peter O’Toole, Harrison Ford, Al Pacino }

pairs = { (The Godfather, Marlon Brando), (House of Wax, Vincent Price), (Casablanca, Humphrey Bogart), (Lawrence of Arabia, Peter O’Toole), (The Fly, Vincent Price) }

Is StarringAgain well-defined?

Yes No

Every domain member is in exactly one pair.

Which codomain items are not in the range of StarringAgain?

Marlon Brando Vincent Price Humphrey Bogart Peter O’Toole Harrison Ford Al Pacino

We can extend our idea of value and image and also think of the reverse.

Image and Preimage

Let $f$ be a function with domain $D$ , range $R$ , and codomain $C$ .
Let $S$ be a subset of the domain $D$ , $S \subseteq D$ .
Let $T$ be a subset of the codomain $C$ , $T \subseteq C$ .

The image of $S$ under $f$ is the collection of function values at all members of $S$ .

$f(S) = \{ f(s) \, | \, s \in S \} \subseteq R \subseteq C$

The preimage of $T$ under $f$ is the collection of domain members whose function values are inside $T$ .

$f^{-1}(T) = \{ d \in D \, | \, f(d) \in T \} \subseteq D$

The preimage is also known as the inverse image.

Define the function $H$ by these sets:

$D = \{ 1, 2, 3, 4, 5, 6, 7, 8 \}$
$R = \{ A, B, C, D \}$
$pairs = \{ (1, A), (2, B), (3, A), (3, C), (4, C), (5, A), (6, B), (7, A), (8, C) \}$

$H(\{ 1, 2 \}) = \{ A, B \}$

$H^{-1}(\{ A, B \}) = \{ 1, 2, 3, 5, 6, 7 \}$

$H^{-1}(\{ C \}) = \{ 3, 4, 8 \}$

$H^{-1}(\{ D \}) = \emptyset$

$H^{-1}(\{ C, D \}) = \{ 3, 4, 8 \}$

$H(\{ 1, 3, 5 \}) = \{ A \}$

Reverse

Mathematics is full of concepts and skills which have a feeling that you are applying something.

For example, multiplication. We can think of multiplication by $3$ as “doing” something. This thought automatically comes with the reverse thought. We might think of reversing multiplication by $3$ as multiplying by $\frac {1}{3}$ and we have alternative exponential notation available, $3^{-1} = \frac {1}{3}$ .

With the addition operation, we think of the opposite of $4$ as $-4 = -1 \cdot 4$ .

Mathematics has many ideas that involve reverse, opposite, backwards, undo, inverse, back up, turn around, etc. In one way or another, $-1$ seems to be included in the notation for this reverse action.

Functions have a feeling of beginning with a domain number and pairing it with a range number. And, naturally, we have the reverse idea.

Our symbol for the reverse of $f$ , is $f^{-1}$ . It begins with range items and pairs them with their domain partners. This reversing of function pairs is certainly a relation, because, well everything is a relation. Whether or not this reverse action results in a function is a topic for later in the course.

The $-1$ superscript has different meanings depending on the context.

Sometimes $f^{-1}(x)$ and $f^{-1}(x)$ do not mean the same thing.

$f^{-1}(x)$ might mean the reciprocal, $\frac {1}{f(x)}$
$f^{-1}(x)$ might mean the inverse function (switch domain and codomain)

Lastly, we should establish when two functions are equal.

Equality

Two functions are equal provided they satisfy all three of the following:

they have the same domain.
they have the same range.
they have the same pairs.

Equality uses the range instead of the codomain, because we are mostly focused on the pairs. However, there are situations where the codomain is important and returns to the picture. We’ll point these situations out when we encounter them.

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more examples can be found by following this link
More Examples of Functions