f(x) "f(x) = ... " is the classic method of composing a function. And over there are various other ways, together you will see!

## Input, Relationship, Output

We will see countless ways come think around functions, however there are constantly three key parts:

The input The relationship The output
input Relationship output
0 × 2 0
1 × 2 2
7 × 2 14
10 × 2 20
...

You are watching: What does input mean in math

... ...

But we space not going to look at specific functions ... ... Instead we will certainly look in ~ the general idea that a function.

## Names

First, the is beneficial to provide a duty a name.

The most typical name is "f", yet we deserve to have other names prefer "g" ... Or even "marmalade" if we want.

But let"s use "f": We say "f of x equals x squared"

what goes into the role is put inside clip () ~ the name of the function:

So f(x) shows us the function is dubbed "f", and "x" walk in

And we normally see what a role does v the input:

f(x) = x2 shows us that duty "f" take away "x" and squares it.

Example: with f(x) = x2:

an entry of 4 becomes an calculation of 16.

In fact we can write f(4) = 16.

## The "x" is simply a Place-Holder!

Don"t gain too concerned around "x", the is simply there to show us whereby the entry goes and what wake up to it.

It might be anything!

So this function:

f(x) = 1 - x + x2

Is the same function as:

f(q) = 1 - q + q2 h(A) = 1 - A + A2 w(θ) = 1 - θ + θ2

The change (x, q, A, etc) is simply there for this reason we understand where to put the values:

f(2) = 1 - 2 + 22 = 3

## Sometimes there is No role Name

Sometimes a duty has no name, and we view something like:

y = x2

But there is still:

an entry (x) a partnership (squaring) and also an output (y)

## Relating

At the height we claimed that a duty was like a machine. Yet a duty doesn"t really have belts or cogs or any moving components - and also it doesn"t actually ruin what us put into it!

A function relates an input come an output.

Saying "f(4) = 16" is prefer saying 4 is somehow regarded 16. Or 4 → 16 Example: this tree grows 20 centimeter every year, therefore the height of the tree is related come its period using the function h:

h(age) = period × 20

So, if the age is 10 years, the elevation is:

h(10) = 10 × 20 = 200 cm

Here space some example values:

ageh(age) = period × 20
00
120
3.264
15300
......

## What types of points Do features Process?

"Numbers" seems an evident answer, yet ... ... which numbers? For example, the tree-height duty h(age) = age×20 makes no sense for an er less than zero. ... That could additionally be letters ("A"→"B"), or ID password ("A6309"→"Pass") or stranger things.

So we require something more powerful, and also that is where sets come in: ### A set is a repertoire of things.

Here room some examples:

Set of also numbers: ..., -4, -2, 0, 2, 4, ...Set of clothes: "hat","shirt",... Set of element numbers: 2, 3, 5, 7, 11, 13, 17, ...Positive multiples the 3 the are less than 10: 3, 6, 9

Each individual point in the set (such together "4" or "hat") is referred to as a member, or element.

So, a function takes elements that a set, and gives back elements the a set.

## A function is Special

But a function has special rules:

It have to work for every feasible input value and it has only one relationship because that each input value

This have the right to be said in one definition: ### Formal an interpretation of a Function

A role relates each element that a setwith precisely one facet of anotherset(possibly the same set).

## The Two vital Things!

 1 "...each element..." means that every element in X is concerned some facet in Y. We say the the role covers X (relates every aspect of it). (But some facets of Y could not be regarded at all, which is fine.)
 2 "...exactly one..." way that a function is single valued. It will not give ago 2 or more results because that the very same input. So "f(2) = 7 or 9" is not right!
 "One-to-many" is not allowed, but "many-to-one" is allowed:  (one-to-many) (many-to-one) This is NOT ok in a function But this is yes in a function

When a relationship does not monitor those two rules then it is not a function ... That is still a relationship, just not a function.

### Example: The relationship x → x2 Could likewise be written as a table:

X: x Y: x2
3 9
1 1
0 0
4 16
-4 16
... ...

It is a function, because:

Every aspect in X is pertained to Y No element in X has actually two or more relationships

So it adheres to the rules.

(Notice how both 4 and also -4 relate to 16, i m sorry is allowed.)

### Example: This partnership is not a function: It is a relationship, but it is not a function, for these reasons:

worth "3" in X has actually no relationship in Y value "4" in X has no relation in Y value "5" is related to an ext than one worth in Y

(But the reality that "6" in Y has no connection does no matter) ## Vertical heat Test

On a graph, the idea of single valued means that no upright line ever crosses an ext than one value.

If it crosses an ext than once it is still a valid curve, yet is not a function.

Some species of attributes have stricter rules, to find out much more you deserve to read Injective, Surjective and Bijective

## Infinitely Many

My examples have just a few values, however functions usually work on sets with infinitely numerous elements.

### Example: y = x3

The output collection "Y" is also all the genuine Numbers

We can"t present ALL the values, so right here are simply a couple of examples:

X: x Y: x3
-2 -8
-0.1 -0.001
0 0
1.1 1.331
3 27
and therefore on... and for this reason on...

## Domain, Codomain and also Range

In our instances above

the set "X" is dubbed the Domain
, the set "Y" is dubbed the Codomain, and also the collection of aspects that obtain pointed to in Y (the yes, really values created by the function) is called the Range.

We have a special web page on Domain, selection and Codomain if you want to understand more.

## So plenty of Names!

Functions have actually been offered in math for a an extremely long time, and also lots of various names and ways of writing functions have come about.

Here space some usual terms you should get acquainted with: ### Example: z = 2u3:

"u" can be referred to as the "independent variable" "z" can be called the "dependent variable" (it depends on the worth of u)

### Example: f(4) = 16:

"4" can be called the "argument""16" can be referred to as the "value that the function"

### Example: h(year) = 20 × year: h() is the function"year" could be dubbed the "argument", or the "variable"a addressed value like "20" deserve to be referred to as a parameter

We often speak to a role "f(x)" once in reality the function is yes, really "f"

## Ordered Pairs

And here is another way to think around functions:

Write the input and output of a role as an "ordered pair", such together (4,16).

They are called ordered pairs due to the fact that the input constantly comes first, and the output second:

(input, output)

So that looks favor this:

( x, f(x) )

Example:

(4,16) way that the function takes in "4" and gives out "16"

### Set of notified Pairs

A duty can then be defined as a set of ordered pairs:

Example: (2,4), (3,5), (7,3) is a role that claims

"2 is pertained to 4", "3 is related to 5" and also "7 is connected 3".

Also, notice that:

the domain is 2,3,7 (the entry values) and also the range is 4,5,3 (the calculation values)

But the function has to it is in single valued, so we likewise say

"if it includes (a, b) and (a, c), climate b must equal c"

Which is simply a method of saying that an input of "a" cannot produce two various results.

Example: (2,4), (2,5), (7,3) is not a duty because 2,4 and 2,5 method that 2 might be pertained to 4 or 5.

In various other words the is no a function because that is not solitary valued ### A benefit of bespeak Pairs

We deserve to graph them...

... Because they are also coordinates!

So a set of coordinates is additionally a duty (if they follow the rule above, the is)

## A duty Can be in Pieces

We can produce functions that behave differently relying on the intake value

### Example: A function with 2 pieces:

when x is less than 0, it gives 5, as soon as x is 0 or an ext it gives x2
-3 Here space some example values: x y 5 -1 5 0 0 2 4 4 16 ... ...

Read much more at Piecewise Functions.

## Explicit vs Implicit

One critical topic: the terms "explicit" and also "implicit".

Explicit is as soon as the function shows us how to go directly from x to y, together as:

y = x3 − 3

When we know x, us can find y

That is the standard y = f(x) stylethat we often work with.

Implicit is once it is not given directly such as:

x2 − 3xy + y3 = 0

When we understand x, how do we uncover y?

It might be difficult (or impossible!) come go straight from x come y.

See more: Meaning Of Number 16 In The Bible, Number 16 Symbolism, 16 Meaning And Numerology

"Implicit" originates from "implied", in other words presented indirectly.

## Conclusion

a role relates inputs to outputs a function takes elements from a collection (the domain) and also relates them to facets in a set (the codomain). Every the outputs (the yes, really values associated to) space together called the rangea role is a special type of relation where: every element in the domain is included, and also any entry produces only one output (not this or that) an input and also its matching output space together called an ordered pairso a role can also be viewed as a set of bespeak pairs
Injective, Surjective and Bijective Domain, variety and Codomain introduction to to adjust Sets Index