Home » [NEW] What is a Function? | each of – NATAVIGUIDES

# [NEW] What is a Function? | each of – NATAVIGUIDES

each of: นี่คือโพสต์ที่เกี่ยวข้องกับหัวข้อนี้

# What is a Function?

A function relates an input to an output.

It is like a machine that has an input and an output.

And the output is related somehow to the input.

f(x)

f(x) = … ” is the classic way of writing a function.
And there are other ways, as you will see!

## Input, Relationship, Output

We will see many ways to think about functions, but there are always three main parts:

• The input
• The relationship
• The output

### Example: “Multiply by 2” is a very simple function.

Here are the three parts:

Input

Output

0
× 2
0

1
× 2
2

7
× 2
14

10
× 2
20

For an input of 50, what is the output?

## Some Examples of Functions

• x2 (squaring) is a function
• x3+1 is also a function
• Sine, Cosine and Tangent are functions used in trigonometry
• and there are lots more!

But we are not going to look at specific functions …
… instead we will look at the general idea of a function.

## Names

First, it is useful to give a function a name.

The most common name is “”, but we can have other names like “” … or even “” if we want.

But let’s use “f”:

We say

what goes into the function is put inside parentheses () after the name of the function:

So shows us the function is called ““, and “” goes in

And we usually see what a function does with the input:

shows us that function “” takes “” and squares it.

Example: with f(x) = x2:

• an input of 4
• becomes an output of 16.

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

## The “x” is Just a Place-Holder!

Don’t get too concerned about “x”, it is just there to show us where the input goes and what happens to it.

It could 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 variable (x, q, A, etc) is just there so we know where to put the values:

f(2) = 1 – 2 + 22 = 3

## Sometimes There is No Function Name

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

y = x2

But there is still:

• an input (x)
• a relationship (squaring)
• and an output (y)

## Relating

At the top we said that a function was like a machine. But a function doesn’t really have belts or cogs or any moving parts – and it doesn’t actually destroy what we put into it!

A function an input to an output.

Saying “f(4) = 16” is like saying 4 is somehow related to 16. Or 4 → 16

Example: this tree grows 20 cm every year, so the height of the tree is to its age using the function :

(age) = age × 20

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

(10) = 10 × 20 = 200 cm

Here are some example values:

age
(age) = age × 20

0
0

1
20

3.2
64

15
300

## What Types of Things Do Functions Process?

seems an obvious answer, but …

which numbers?

For example, the tree-height function (age) = age×20 makes no sense for an age less than zero.

… it could also be letters (“A”→”B”), or ID codes (“A6309″→”Pass”) or stranger things.

So we need something more powerful, and that is where sets come in:

### A set is a collection of things.

Here are some examples:

• Set of even numbers: {…, -4, -2, 0, 2, 4, …}
• Set of clothes: {“hat”,”shirt”,…}
• Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, …}
• Positive multiples of 3 that are less than 10: {3, 6, 9}

Each individual thing in the set (such as “4” or “hat”) is called a member, or element.

So, a function takes elements of a set, and gives back elements of a set.

## A Function is Special

But a function has special rules:

• It must work for every possible input value
• And it has only one relationship for each input value
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This can be said in one definition:

### Formal Definition of a Function

A function relates each element of a set
with exactly one element of another
set
(possibly the same set).

## The Two Important Things!

1.

“…each element…” means that every element in X is related to some element in Y.

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

(But some elements of Y might not be related to at all, which is fine.)

2.

“…exactly one…” means that a function is . It will not give back 2 or more results for the 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 OK in a function

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

### Example: The relationship x → x2

Could also 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 element in X is related to Y
• No element in X has two or more relationships

So it follows the rules.

(Notice how both 4 and -4 relate to 16, which is allowed.)

### Example: This relationship is not a function:

It is a relationship, but it is not a function, for these reasons:

• Value “3” in X has no relation in Y
• Value “4” in X has no relation in Y
• Value “5” is related to more than one value in Y

(But the fact that “6” in Y has no relationship does not matter)

## Vertical Line Test

On a graph, the idea of single valued means that no vertical line ever crosses more than one value.

If it crosses more than once it is still a valid curve, but is not a function.

Some types of functions have stricter rules, to find out more you can read Injective, Surjective and Bijective

## Infinitely Many

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

### Example: y = x3

• The input set “X” is all Real Numbers
• The output set “Y” is also all the Real Numbers

We can’t show ALL the values, so here are just a few examples:

X: x
Y: x3

-2
-8

-0.1
-0.001

0
0

1.1
1.331

3
27

and so on…
and so on…

## Domain, Codomain and Range

In our examples above

• the set “X” is called the Domain,
• the set “Y” is called the Codomain, and
• the set of elements that get pointed to in Y (the actual values produced by the function) is called the Range.

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

## So Many Names!

Functions have been used in mathematics for a very long time, and lots of different names and ways of writing functions have come about.

Here are some common terms you should get familiar with:

### Example: z = 2u3:

• “u” could be called the “independent variable”
• “z” could be called the “dependent variable” (it depends on the value of u)

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

• “4” could be called the “argument”
• “16” could be called the “value of the function”

### Example: h(year) = 20 × year:

• h() is the function
• “year” could be called the “argument”, or the “variable”
• a fixed value like “20” can be called a parameter

We often call a function “f(x)” when in fact the function is really “f”

## Ordered Pairs

And here is another way to think about functions:

Write the input and output of a function as an “ordered pair”, such as (4,16).

They are called ordered pairs because the input always comes first, and the output second:

(input, output)

So it looks like this:

( x, f(x) )

Example:

(4,16) means that the function takes in “4” and gives out “16”

### Set of Ordered Pairs

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

Example: {(2,4), (3,5), (7,3)} is a function that says

“2 is related to 4”, “3 is related to 5” and “7 is related 3”.

Also, notice that:

• the domain is {2,3,7} (the input values)
• and the range is {4,5,3} (the output values)

But the function has to be single valued, so we also say

“if it contains (a, b) and (a, c), then b must equal c”

Which is just a way of saying that an input of “a” cannot produce two different results.

Example: {(2,4), (2,5), (7,3)} is not a function because {2,4} and {2,5} means that 2 could be related to 4 or 5.

In other words it is not a function because it is not single valued

### A Benefit of Ordered Pairs

We can graph them…

… because they are also coordinates!

So a set of coordinates is also a function (if they follow
the rules above, that is)

## A Function Can be in Pieces

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

### Example: A function with two pieces:

• when x is less than 0, it gives 5,
• when x is 0 or more it gives x2

Here are some example values:

x
y

-3
5

-1
5

0
0

2
4

4
16

Read more at Piecewise Functions.

## Explicit vs Implicit

One last topic: the terms “explicit” and “implicit”.

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

y = x3 − 3

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That is the classic y = f(x) style that we often work with.

Implicit is when it is not given directly such as:

x2 − 3xy + y3 = 0

It may be hard (or impossible!) to go directly from x to y.

“Implicit” comes from “implied”, in other words shown indirectly.

## Graphing

• The Function Grapher can only handle explicit functions,
• The Equation Grapher can handle both types (but takes a little longer, and sometimes gets it wrong).

## Conclusion

• a function relates inputs to outputs
• a function takes elements from a set (the domain) and relates them to elements in a set (the codomain).
• all the outputs (the actual values related to) are together called the range
• a function is a special type of relation where:
• every element in the domain is included, and
• any input produces only one output (not this or that)
• an input and its matching output are together called an ordered pair
• so a function can also be seen as a set of ordered pairs

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## [NEW] 5 jQuery.each() Function Examples | each of – NATAVIGUIDES

This is an extensive overview of the `jQuery.each()` function — one of jQuery’s most important and most used functions. In this article, we’ll find out why and take a look at how you can use it.

## What is jQuery.each()

jQuery’s each() function is used to loop through each element of the target jQuery object — an object that contains one or more DOM elements, and exposes all jQuery functions. It’s very useful for multi-element DOM manipulation, as well as iterating over arbitrary arrays and object properties.

In addition to this function, jQuery provides a helper function with the same name that can be called without having previously selected or created any DOM elements.

## jQuery.each() Syntax

Let’s see the different modes in action.

The following example selects every `<div>` element on a web page and outputs the index and the ID of each of them:

``````
\$('div').each(function(index, value) {
console.log(`div\${index}: \${this.id}`);
});
``````

A possible output would be:

``````div0:header
div1:main
div2:footer
``````

This version uses jQuery’s `\$(selector).each()` function, as opposed to the utility function.

The next example shows the use of the utility function. In this case the object to loop over is given as the first argument. In this example, we’ll show how to loop over an array:

``````
const arr = [
'one',
'two',
'three',
'four',
'five'
];

\$.each(arr, function(index, value) {
console.log(value);

return (value !== 'three');
});

``````

In the last example, we want to demonstrate how to iterate over the properties of an object:

``````
const obj = {
one: 1,
two: 2,
three: 3,
four: 4,
five: 5
};

\$.each(obj, function(key, value) {
console.log(value);
});

``````

This all boils down to providing a proper callback. The callback’s context, `this`, will be equal to its second argument, which is the current value. However, since the context will always be an object, primitive values have to be wrapped:

``````\$.each({ one: 1, two: 2 } , function(key, value) {
console.log(this);
});

``````

`

This means that there’s no strict equality between the value and the context.

``````\$.each({ one: 1 } , function(key, value) {
console.log(this == value);
console.log(this === value);
});

``````

`

The first argument is the current index, which is either a number (for arrays) or string (for objects).

## 1. Basic jQuery.each() Function Example

Let’s see how the jQuery.each() function helps us in conjunction with a jQuery object. The first example selects all the `a` elements in the page and outputs their `href` attribute:

``````\$('a').each(function(index, value){
console.log(this.href);
});
``````

The second example outputs every external `href` on the web page (assuming the HTTP(S) protocol only):

``````\$('a').each(function(index, value){
const link = this.href;

}
});
``````

Let’s say we had the following links on the page:

``````<a href="https://www.sitepoint.com/">SitePoint</a>
<a href="https://developer.mozilla.org">MDN web docs</a>
<a href="http://example.com/">Example Domain</a>
``````

The second example would output:

``````https://www.sitepoint.com/
https://developer.mozilla.org/
http://example.com/
``````

We should note that DOM elements from a jQuery object are in their “native” form inside the callback passed to `jQuery.each()`. The reason is that jQuery is in fact just a wrapper around an array of DOM elements. By using `jQuery.each()`, this array is iterated in the same way as an ordinary array would be. Therefore, we don’t get wrapped elements out of the box.

With reference to our second example, this means we can get an element’s `href` attribute by writing `this.href`. If we wanted to use jQuery’s `attr()` method, we would need to re-wrap the element like so: `\$(this).attr('href')`.

## 2. jQuery.each() Array Example

Let’s have another look at how an ordinary array can be handled:

``````const numbers = [1, 2, 3, 4, 5];
\$.each(numbers, function(index, value){
console.log(`\${index}: \${value}`);
});
``````

This snippet outputs:

``````:1
1:2
2:3
3:4
4:5
``````

Nothing special here. An array features numeric indices, so we obtain numbers starting from 0 and going up to N-1, where N is the number of elements in the array.

## 3. jQuery.each() JSON Example

We may have more complicated data structures, such as arrays in arrays, objects in objects, arrays in objects, or objects in arrays. Let’s see how `jQuery.each()` can help us in such scenarios:

``````const colors = [
{ 'red': '#f00' },
{ 'green': '#0f0' },
{ 'blue': '#00f' }
];

\$.each(colors, function() {
\$.each(this, function(name, value) {
console.log(`\${name} = \${value}`);
});
});
``````

This example outputs:

``````red =
green =
blue =
``````

We handle the nested structure with a nested call to `jQuery.each()`. The outer call handles the array of the variable `colors`; the inner call handles the individual objects. In this example each object has only one key, but in general, any number could be tackled with this code.

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## 4. jQuery.each() Class Example

This example shows how to loop through each element with assigned class `productDescription` given in the HTML below:

``````<div class="productDescription">Red</div>
<div>Pink</div>
<div class="productDescription">Orange</div>
<div class="generalDescription">Teal</div>
<div class="productDescription">Green</div>
``````

We use the `each()` helper instead of the `each()` method on the selector.

``````\$.each(\$('.productDescription'), function(index, value) {
console.log(index + ':' + \$(value).text());
});
``````

In this case, the output is:

``````:Red
1:Orange
2:Green
``````

We don’t have to include index and value. These are just parameters that help determine which DOM element we’re currently iterating on. Furthermore, in this scenario we can also use the more convenient `each` method. We can write it like this:

``````\$('.productDescription').each(function() {
console.log(\$(this).text());
});
``````

And we’ll obtain this on the console:

``````Red
Orange
Green
``````

Note that we’re wrapping the DOM element in a new jQuery instance, so that we can use jQuery’s `text()` method to obtain the element’s text content.

## 5. jQuery.each() Delay Example

In the next example, when the user clicks the element with the ID `5demo` all list items will be set to orange immediately.

``````<ul id="5demo">
<li>One</li>
<li>Two</li>
<li>Three</li>
<li>Four</li>
<li>Five</li>
</ul>
``````

After an index-dependent delay (0, 200, 400, … milliseconds) we fade out the element:

``````\$('#5demo').on('click', function(e) {
\$('li').each(function(index) {
\$(this).css('background-color', 'orange')
.delay(index * 200)
});

e.preventDefault();
});
``````

## Conclusion

In this post, we’ve demonstrated how to use the `jQuery.each()` function to iterate over DOM elements, arrays and objects. It’s a powerful and time-saving little function that developers should have in their toolkits.

And if jQuery isn’t your thing, you might want to look at using JavaScript’s native Object.keys() and Array.prototype.forEach() methods. There are also libraries like foreach which let you iterate over the key value pairs of either an array-like object or a dictionary-like object.

Remember: `\$.each()` and `\$(selector).each()` are two different methods defined in two different ways.

This popular article was updated in 2020 to reflect current best practices and to update the conclusion’s advice on native solutions using modern JavaScript. For more in-depth JavaScript knowledge, read our book, JavaScript: Novice to Ninja, 2nd Edition.

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