"f(x) = ...

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" is the classic means of writing a function.And there are other ways, together you will see!

Input, Relationship, Output

We will certainly see many ways to think around functions, yet there are constantly three key parts:

The entry The partnership The calculation

But we space not going to look at specific functions ...... Instead we will look at the basic idea of a function.


First, it is useful to offer a role a name.

The most common name is "f", yet we have the right to have other names prefer "g" ... Or even "marmalade" if us want.

But let"s usage "f":


We say "f the x equates to x squared"

what walk into the role is placed inside clip () after the surname of the function:

So f(x) mirrors us the function is called "f", and also "x" goes in

And we typically see what a duty does through the input:

f(x) = x2 reflects us that role "f" takes "x" and squares it.

Example: through f(x) = x2:

an input of 4 i do not care an calculation of 16.

In fact we deserve to write f(4) = 16.

The "x" is simply a Place-Holder!

Don"t obtain too concerned around "x", that is just there to show us where the entry goes and also what wake up to it.

It can be anything!

So this function:

f(x) = 1 - x + x2

Is the same role as:

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

The change (x, q, A, etc) is just there so we understand where to put the values:

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

Sometimes there is No role Name

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

y = x2

But there is still:

an intake (x) a relationship (squaring) and also an calculation (y)


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

A function relates an input come an output.

Saying "f(4) = 16" is choose saying 4 is somehow related to 16. Or 4 → 16


Example: this tree grows 20 centimeter every year, therefore the elevation of the tree is related come its age using the duty h:

h(age) = period × 20

So, if the period is 10 years, the height is:

h(10) = 10 × 20 = 200 cm

Here room some example values:

age h(age) = age × 20
0 0
1 20
3.2 64
15 300
... ...

What species of things Do features Process?

"Numbers" seems an evident answer, however ...

... Which numbers?

For example, the tree-height duty h(age) = age×20 provides no sense for period less 보다 zero.

... It could likewise be letters ("A"→"B"), or ID codes ("A6309"→"Pass") or stranger things.

So we need something much more powerful, and also that is wherein sets come in:


A collection is a arsenal of things.

Here space some examples:

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

Each separation, personal, instance thing in the set (such together "4" or "hat") is referred to as a member, or element.

So, a role takes elements the a set, and also gives ago elements that a set.

A role is Special

But a duty has special rules:

It should work because that every possible input value and it has actually only one relationship because that each input value

This have the right to be said in one definition:


Formal meaning of a Function

A duty relates each element the a setwith specifically one facet of an additional set(possibly the very same set).

The Two crucial Things!


"...each element..." way that every facet in X is concerned some aspect in Y.

We say that the role covers X (relates every element of it).

(But some facets of Y can not be concerned at all, which is fine.)


"...exactly one..." method that a function is single valued. It will certainly not give ago 2 or much more results because that the exact same input.

So "f(2) = 7 or 9" is not right!

"One-to-many" is not allowed, however "many-to-one" is allowed:

(one-to-many) (many-to-one)
This is NOT yes in a function But this is ok in a function

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

Example: The relationship x → x2


Could additionally be created 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 concerned Y No facet in X has two or much more relationships

So it adheres to the rules.

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

Example: This connection is not a function:


It is a relationship, however it is not a function, because that these reasons:

value "3" in X has no relation in Y worth "4" in X has no relation in Y value "5" is associated to much more than one worth in Y

(But the truth that "6" in Y has actually no relationship does not matter)


Vertical line Test

On a graph, the idea of single valued method that no upright line ever crosses much more than one value.

If the crosses an ext than once it is tho a valid curve, however is not a function.

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

Infinitely Many

My instances have simply a couple of values, but functions usually work-related on sets through infinitely numerous elements.

Example: y = x3

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

We can"t present ALL the values, so below are simply a few examples:

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

Domain, Codomain and Range

In our instances above

the set "X" is referred to as the Domain, the collection "Y" is referred to as the Codomain, and also the set of facets that get pointed to in Y (the yes, really values developed by the function) is referred to as the Range.

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

So many Names!

Functions have been provided in math for a very long time, and lots of different names and also ways the writing attributes have come about.

Here space some usual terms you should get familiar with:


Example: z = 2u3:

"u" might be dubbed the "independent variable" "z" can be called the "dependent variable" (it depends on the worth of u)

Example: f(4) = 16:

"4" could be dubbed the "argument" "16" can be called the "value that the function"

Example: h(year) = 20 × year:


h() is the duty "year" could be called the "argument", or the "variable" a solved value prefer "20" have the right to be dubbed a parameter

We often call a duty "f(x)" when in reality the duty is really "f"

Ordered Pairs

And right here is another means to think around functions:

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

They are referred to as ordered pairs since the input constantly comes first, and also the calculation second:

(input, output)

So it looks like this:

( x, f(x) )


(4,16) method that the role takes in "4" and gives out "16"

Set of notified Pairs

A role can then be identified as a set of ordered pairs:

Example: (2,4), (3,5), (7,3) is a duty that says

"2 is concerned 4", "3 is concerned 5" and also "7 is connected 3".

Also, notice that:

the domain is 2,3,7 (the intake values) and the variety is 4,5,3 (the output values)

But the role has to be single valued, for this reason we also say

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

Which is just a way of saying the an intake of "a" cannot create two different results.

Example: (2,4), (2,5), (7,3) is not a function because 2,4 and also 2,5 way that 2 can be regarded 4 or 5.

In other words that is no a role because it is not solitary valued


A benefit of notified Pairs

We can graph them...

... Due to the fact that they are likewise coordinates!

So a set of collaborates is additionally a duty (if they follow the rules above, that is)

A role Can be in Pieces

We can create functions that behave differently depending on the input value

Example: A duty with 2 pieces:

once x is less than 0, it offers 5, once x is 0 or much more it provides x2
Here room some instance values: x y
-1 5
0 0
2 4
4 16
... ...

Read 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 just how to go directly from x to y, together as:

y = x3 − 3

When we understand x, us can find y

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

Implicit is as soon as it is not given directly such as:

x2 − 3xy + y3 = 0

When we recognize x, just how do we uncover y?

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

See more: How Many Cups Are In 3 Gallons To Cups, How Many Cups

"Implicit" comes from "implied", in various other words shown indirectly.



a duty relates inputs come outputs a function takes elements from a collection (the domain) and relates them to facets in a collection (the codomain). Every the outputs (the yes, really values connected to) space together called the range a duty is a special type of relationship where: every element in the domain is included, and also any input produces only one output (not this or that) an input and also its equivalent output room together dubbed an ordered pair therefore a duty can additionally be seen as a set of ordered pairs
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Injective, Surjective and Bijective Domain, range and Codomain introduction to to adjust Sets Index