f(x) | "f(x) = ... " is the classic method of composing a function. |

## Input, Relationship, Output

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

The input The relationship The outputinput 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**:

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 + θ2The change (x, q, A, etc) is simply there for this reason we understand where to put the values:

f(**2**) = 1 - **2** + **2**2 = 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:

age

**h(age) = period × 20**

0 | 0 |

1 | 20 |

3.2 | 64 |

15 | 300 |

... | ... |

## What types of points Do features Process?

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

... For example, the tree-height duty | |

... 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
So, a function takes ## A function is SpecialBut a function has 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 FunctionA role relates ## The Two vital Things!
When a relationship does ## Example: The relationship x → x2Could likewise be written as a table: X: x Y: x2
So it adheres to the rules. (Notice how both ## Example: This partnership is |

-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 parameterWe 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**.

## Graphing

## 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

**range**a 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 pair**so 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