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Straight Line Graph/Graphing Linear Inequalities Polynomial Factoring Solving Linear Inequalities Intro Circle Geometry, Number Pi Polynomial Definition Straight Line Graph/Graphing Linear Inequalities Integers, Adding/SubtractingIntroduction to Functions

This introduction to Functions page looks to give a solid understanding, explaining what functions in
Math are, and how they work.

A **function** defines a relationship between numbers.

It is a rule that assigns, for each number/element of a set, say * x* of set

A unique element of another set, say

This can be illustrated to give a clearer idea, such as in the image below.

Most of the time, the notation for functions in Math is * f(x)* =

Where

The letter

The letters involved don’t always have to be

For example

Also,

Sometimes a function can also be denoted

A function from a set

Domain and Range

Consider the function

On a set

Set **X** is called the domain, which is the set containing the values going into the function.

Set **Y** is called the range or image, this set contains values coming out of the function.

Sometimes set **Y** can contain some extra values, that aren’t related to any values in set
**X** by a specific function.

When this is the case, set **Y** is referred to as the co-domain, such a situation is
illustrated below.

As well as being written as * f(x)* =

A basic function can also be written as

As it's the

Injective/Surjective

A function ** f: X → Y**,

is INJECTIVE if it takes different elements in set

For example, with the previous sets

One element of set

The two set’s pictured below also illustrate an injective function.

These sets are INJECTIVE.

Even though  2 elements in set **Y** are not assigned to an element in set **X**,
it’s still one to one correspondence for the elements that are assigned.

-----------------------

A function is SURJECTIVE if every element in set **Y**, can be written as * f(x)*,
for an element of set

Take the function

A set

Also a set

Every element in **Y** can be written as a square of an element in set **X**,

so here the function * f(x)* =

If a function is BOTH surjective and injective with regards to its domain and co-domain, then it is a BIJECTIVE function.

A function can also be classed as Explicit or Implicit.

An EXPLICIT function is a simpler function, like the ones seen earlier in this page.

Where the * y* value can be obtained directly from the

Such as

An IMPLICIT function is one of the form:

Implicit functions are usually more complicated to deal with.