1. Functions CSRU1400, Spring 2008 Ellen Zhang CSRU1400 Spring 2008

2. Outline What is a function ? Web project: part 1 and 2
Property of functions: onto, one-to-one
Pigeonhole principle
Inverse function & function composition
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Ellen Zhang

3. Remember binary relations ?
A binary relation is a mapping between two sets as defined by a rule.
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4. Some relations are special
We call these special relations “functions” and you have probably met them before.
A function is a way of transforming one set of things (usually numbers) into another set of things (also usually numbers).
For example:
Fahrenheit to Celsius Conversion (link)
[℃] = ([°F] − 32) × 5⁄9
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Ellen Zhang

5. More examples of functions
A function that maps a student to his/her major (or age, height, …)
GDP of a country: maps the country to GDP (Gross Domestic Production)
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6. Components of All Functions
Every functions must have
Name, typically we use a letter like f, g, or h to name a function
Domain, a set of values
Codomain, a set of values
Rule: maps values in the domain to values in the codomain
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7. Terminology f : A B
This means that the function f maps values in the set A (the domain) to the values in the set B (the codomain)
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8. Simple rules
We can specify the rules of functions by enumerating ordered pairs that represents the mapping.
Name: AddFour
Domain: {1, 2, 3}
Codomain: {5, 6, 7, 8}
Rule: {(1, 5), (2, 6), (3, 7)}
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9. Using mathematic formula
Rules can also be specified using formula
Functions mapping numbers to numbers
f(a) = a + 4, “f of a equals a plus 4”
g(b) = b * b + 2, “g of b equals b times b plus 2”
h(c) = 5, “h of c equals 5”
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10. Rules continued
When we see rules we often ask what their value might be when given concrete values.
Take the formula from the previous page
f(a) = a+4
What is its value when a equals 7?
Answer: 11
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11. More Rules
To abbreviate this questions I might just ask what is the f(7)?
This means substitute the 7 for the a and solve the equation on the right hand side.
Try out the following formulas and related questions on the next slide.
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12. Formula Problems f(x) = 2x + 3 g(x) = 7 What is the f(5)?
What is the g(3)?
What is the g(7)?
What is the g(-10)? 13 19 -5 7 7 7
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13. So what is a function really?
A function has three components: domain, codomain and rule; and
you can take every value in the domain, use the rule to map the value into a single value that is in the Codomain
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14. Definition of function
A function F from A to B is a binary relation with domain A and codomain B, where for every , there is exactly one element for which
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15. But that makes no sense? Let’s look at an example. Is this a function?
Suppose I define my domain to be {1, 2, 3}
And I define my Codomain to be {5, 6, 7, 8}
And my formula is f(x) =x + 5
Is this a function?
To find out, be very meticulous and walk through each value of the domain
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16. Let me try the value 1. f(1) = 1+ 5 = 6
Domain = {1, 2, 3} Codomain = {5, 6, 7, 8} f(x) = x + 5
Let me try the value 1. f(1) = 1+ 5 = 6
6 is in my Codomain. So far it’s working
Let me try the value 2. f(2) = = 7
7 is in my Codomain. It is still working
Let me try the value 3. f(3) = = 8
8 is in my Codomain. It is still working
I have tried all values in my domain, and they are all mapped to a value in codomain.
Therefore this is a function
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17. Taking 1 from the domain, works Taking 2 from the domain, works
Domain = {1, 2, 3}, Codomain = {5, 6, 7}, f(x) = x + 5
Almost identical.
Taking 1 from the domain, works
Taking 2 from the domain, works
Taking 3 from the domain, fails.
Why? The rule f(x) maps 3 to value 8, which is not in my Codomain. Therefore this example is not a function.
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18. Choose 0. f(x) = 5 … it’s in the Codomain
Domain = Z (all integers) Codomain = Z (all integers)
f(x) = x + 5
Ok, begin the same way (take values from the domain and put them in the formula)
Choose 0. f(x) = 5 … it’s in the Codomain
Choose 1. f(1) = 6 … it’s in the Codomain
Choose -1. f(-1) = it’s in the Codomain
But we can’t do this forever
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19. Dealing with infinite sets
If the domain is infinite, you can’t try all values in the domain
So you need to look for values that might not work and try those.
If you can’t find any domain values that don’t work, can you make an argument that all the domain values do work ?
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20. Domain = Z (all integers) Codomain = Z (all integers)
f(x) = x + 5
Hand-waving argument
“Regardless of what integer I take from the domain, I can add 5 to that number and still have a value in the Codomain.”
Once you convince yourself of this argument then you know it is a function
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21. It always works… so it is a function
Domain = Z (all integers) Codomain = {4, 5, 6} f(x) = 6
Ok choose some values
Choose 0: f(0) = 6 … it works
Choose 1: f(1) = 6 … it works
Choose -1: f(-1) = 6 … it works
It always works… so it is a function
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22. Choose some values so it’s not a function
Domain = N (natural numbers, i.e., 0,1,2,…)
Codomain = N
f(x) = x-1
Choose some values
Choose 0: f(0) = -1 … -1 is not in the codomain, it doesn’t work
so it’s not a function
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23. We have seen many functions so far …
Are the following functions ? What are their domains and codomains?
f(A)=#A-1
f(x) = (x>0)
And
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24. Functions with multiple variables
Example:
f(x,y)=x-y, where x takes integer value, and y takes integer value
f maps an ordered pair of integers, i.e., x and y, to their difference (x-y), which is also an integer
What’s the domain ?
The set of ordered pairs of integers …
In mathematical notation: , the Cartesian product of Z
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25. More examples What are the domain, codomains ? f(x,y)=x>y 2.
3. Union(A,B)=
4. SizeofProduct(A,B) = #A * #B
5. NumOfSubsets (A) = 2#A
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26. Web Projects: develop your own web pages and make them interesting/useful …
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27. What is a Computer ? Computer is a device capable of hardware devices
Performing computations
Making logical decisions
Works billions of times faster than human beings
hardware devices
Keyboard, screen (monitor)
Disks, Memory
Processing Units
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28. What is Software ? Computer Programs are called software
sets of instructions that process data, e.g.,
Add two numbers
If some condition is true, then go to execute a certain instruction
guide computer through orderly sets of actions specified by computer programmers
Programmers write instructions that comprise software in various programming languages
Java, C ++(CS1)
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29. The Internet
A "network of networks" that consists of millions of smaller domestic, academic, business, and government networks.
Worldwide, publicly accessible
Mixing computing and communications technologies
Carrying information and services, such as electronic mail, online chat, file transfer, and the interlinked Web pages and other documents of the World Wide Web.
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30. The World Wide Web Introduced in 1990 by Tim Berners-Lee
A system of interlinked, hypertext documents accessed via the Internet.
With a web browser, a user views web pages that may contain text, images, video, audio, and other multimedia, and navigates between them using hyperlinks.
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31. What happens behind the scene ?
A web browser (client) lets a user request a resource.
A web server takes the client request and sends something back to the client.
Clients and servers know HTML (HyperText Markup Language), the agreed-upon standard for web pages
Client
Request
Response
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32. What is Web Browser ? A Web browser is a software that
enables a user to display and interact with the Web’s rich multimedia content such as text, images, and other information.
The Web could be the World Wide Web, a local area network, or a web page on your own machine.
Microsoft Internet Explorer, Mozilla Firefox, Safari or Netscape Navigator
The appearance of a Web page may differ between browsers.
Netscape Browser will Die in February 2008
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33. URL & Hyperlinks URL (Uniform/Universal Resource Locator) Hyperlinks
Web page address – typing in Address field
HTTP (HyperText Transfer Protocol)
Protocol for transferring data over the Internet
HTTPS (Secure HyperText Transfer Protocol)
Protocol for transferring encrypted data over the Internet.
Hyperlinks
Graphical or textual elements
Click to link to another Web page
Loads new page into browser window
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34. Our computing environment: server
The server is storm.cis.fordham.edu
An account with the server has been created for you (if you have requested for it)
You have an account name and password, also a file directory, account, a URL
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35. The client: OpenMail
Our client software is OpenMail:
Using OpenMail, you can
Check s (WebMail button)
View/Manipulate your directory (WebDisk)
Configure your account (e.g., change passwd, Pref)
Detailed instruction at the course web site…
Create, edit a file, upload a file …
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36. Steps to build your web pages
URL for your web page with storm is:
When someone (anyone) enters above URL into web browser, browser will try to fetch & display
index.html file located under ~<your_account>/public_html/, at storm
Therefore, you need to create a file named index.html under public_html directory
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37. Build you home page …
Your index.html have to follow certain standard, HTML or XHTML
Start with the simple example
copy and paste it to your editor
save and preview it
Experiment by adding one thing a time
Borrow someone else’s good ideas
When you see a cool web page, select to view its source in the browser
Copy and Paste to your editor
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38. First HTML Example (1) <html> Basic HTML Elements: <head>
<title>My first Web Page</title>
</head> 
<body bgcolor = "white"> 
<h2> My first Web Page</h2>
<p>I am so happy to be at Fordham!
</body>
</html>
head element
Head section: Title of the document and Style sheets and scripts
body element
Body section: Page’s content the browser displays
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39. What is HTML Element ? A name and value pair Example:
Every element starts with Start tag and ends with End tag, with the displayed content in between them.
Example: <html>. . .</html>, <head>. . .</head>.
Start tag may have attributes (provide additional information about an element)
A name and value pair
Example:
<body bgcolor = "white"> 
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40. Headers Elements Six headers ( header elements): h1 ~ h6
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41. CSRU1400 Spring 2008
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42. Adding HyperLink
References other sources such as HTML documents and images
Both text and images can act as hyperlinks
Created using the a (anchor) element: <a> </a>
Attribute href specifies the location of a linked resource :
href = “
Link to addresses:
href =
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43. links.html (1 of 2)
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44. Adding Images Element (1)
img element with attributes:
src attribute : Specifies the location of the image file
width and height attributes: Pixels (“picture elements”)
alt attribute : the text will be displayed if the browser could not display the image.
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45. Adding Images Element (2)
Image elements are empty, i.e., nothing goes between start tag and end tag
<img src=“1.jpg” height=“238” width=“183”></img>
Shorthand: terminated by character / inside the closing right angle bracket (>)
<img src=“1.jpg” height=“238” width=“183” />
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46. picture.html (1 of 1)
Ellen Zhang
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47. CSRU1400 Spring 2008
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48. Image as Link Use an image as a link
<a href=
<img src= "yahoo.gif" width="232" height = "44" />
</a>
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49. Internal Linking
Enables user to jump between locations in the same document
First, create an internal hyperlink destination by setting attribute id of an element
Second, create an internal link to this element.
Example:
<h1 id=“bugs”>Bugs </h1> :
<a href=“#bugs”>Go to Bugs</a>
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50. links.html (1 of 3)
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51. links.html (3 of 3)
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52. Unordered Lists Unordered list element ul
Creates a list in which each item begins with a bullet symbol (called a disc)
li (list item): Entry in an unordered list
Format:
<ul>
<li> first item </li>
<li> second item </li>
</ul>
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53. Nested and Ordered Lists
Nested lists:
Represent hierarchical relationships
Ordered lists (ol)
Creates a list in which each item begins with a number
Format
<ol>
<li> first item </li>
<li> second item </li>
</ol>
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54. list.html (1 of 3)
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55. list.html (2 of 3)
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56. list.html (3 of 3)
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57. CSRU1400 Spring 2008
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58. We need to do some simple programming …
So far, everything is static, i.e., the contents of the web page is fixed…
How to make the content adapts to the viewer, and the time ? Or make your web page interact with viewer ?
We need to do some simple programming …
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Ellen Zhang

59. JavaScript JavaScript scripting language
Enhances functionality and appearance
Web browser contains a JavaScript interpreter.
JavaScript is an Object Oriented Programming (OOP) language.
Everything is an object
Each object is associated with some attributes (data) and behaviors (methods).
A statement should end with a semicolon (;).
JavaScript is case sensitive.
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60. Simple JavaScript Program
Inline scripting
Written in the <body> or <head> of a document
<script> tag
JavaScript Single-line comment symbol: //
Example: <script type=“text/javascript”>
<! - -
script code here
// - - >
</script>
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61. Programming with JavaScript (1)
document object: the HTML document the browser is currently displaying
writeln method writes a line in the document
document.writeln(“<h1>Welcome</h1>”);
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62. Example One <script language="javascript“
type="text/javascript">
<!--
document.write(“<p>Last updated on " +
document.lastModified + “</p>”);
-->
</script>
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63. Programming with JavaScript (2)
window object : a browser window.
A Window object is created automatically with every instance of a <body> or <frameset> tag
alert method creates a Dialog box
window.alert(“welcome”);
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64. welcome.html (1 of 1)
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65. welcome4.html 1 of 1
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66. Variables Variables are used to store data.
Keyword : var
A variable's value can change during the script.
You can refer to a variable by name to see its value or to change its value.
Name of a variable : a series of characters
letters, digits, underscores( _ ) and dollar signs ($)
no white space
not begin with a digit
not a keyword.
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67. Use Variables Declare a variable: Assign a value to a variable:
var name;
var size;
var name, size;
Assign a value to a variable:
name = “Lisa”;
size = 20;
Combine two steps:
var name = “Lisa”;
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68. Dynamic Welcome Page
A script can adapt the content based on input from the user or other variables
prompt method of window object
window.prompt(“Please enter your name”, “Lisa”);
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69. welcome5.html (1 of 2)
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70. CSRU1400 Spring 2008
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71. This is the prompt to the user.
When the user clicks OK, the value typed by the user is returned to the program as a string.
This is the prompt to the user.
This is the default value that appears when the dialog opens.
This is the text field in which the user types the value.
Fig. 7.7 Prompt dialog displayed by the window object’s prompt method.
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72. Arithmetic JavaScript performs arithmetic calculations.
Arithmetic operators:
Addition: numberOne + numberTwo
Subtraction: numberOne – 5
Multiplication: numberOne * numberTwo
Division: numberTwo / 6
Rules of operator precedence
a * ( b – c )
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73. Adding Integers Prompt user for two integers and calculate the sum
parseInt method
Converts its string argument to an integer
parseFloat method
Converts its string argument to a float
Assignment statement
sum = number1 + number2;
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74. Addition.html (1 of 2)
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75. Addition.html (2 of 2)
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76. CSRU1400 Spring 2008
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77. Part II of web project: a degree converter
After finish part I , extend your website
Create a converter.html file
Prompt viewer to input temperature in Fahrenheit
Show the degree in Celsius (see formula in slide 4)
Add a hyperlink in your main file, i.e., index.html, to the converter.html
Here are useful links for doing arithmetic in JavaScript
You are welcomed to experiment more…
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78. Outline Definition of function Property of functions Inverse function
Onto
One-to-one
Pigeonhole principle
Inverse function
Function composition
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79. Function f: S→T s t f(s)=t f maps s to t t is called the image of s.
Codomain T
Domain S s t f
f(s)=t
f maps s to t
t is called the image of s.
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80. Properties of Functions
Two interesting properties of functions
A function can be onto
A function can be one-to-one
Note: If the relation is not a function than these properties are irrelevant
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81. Onto Functions
A function is onto if every value in the codomain is the image of some value in the domain.
Formally, function f: S→T is onto if
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82. Onto example First you have to figure out if it is a function or not.
Domain = {1, 2, 3, 4} Codomain = {11, 12, 13, 14} f(x) = x + 10
First you have to figure out if it is a function or not.
Choose 1: f(1) = 11
Choose 2: f(2) = 12
Choose 3: f(3) = 13
Choose 4: f(4) = 14
So it is a function
Is it onto?
Yes. Because we covered all of the Codomain values, i.e., every value of codomain is an image of some values in the domain.
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83. Determine whether it is a function
Domain = {1, 2, 3,} Codomain = {0, 1, 2, 3} f(x) = x -1
Determine whether it is a function
Choose 1: f(1) = 0
Choose 2: f(2) = 1
Choose 3: f(3) = 2
So it is a function.
Is it onto?
No, we never arrived at the value 3 which is in the Codomain
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84. Dealing with infinite sets (1)
f: R→R with the rule f(x)=x2+1
Is it a function?
Is it an onto function ?
For every
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85. Dealing with infinite sets (2)
g: Q+→Q+ with the rule
f: R→R with the rule f(x)=x2+4x+1
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86. One-to-One
A function is one-to-one if each value in the Codomain that is reached can only be reached by one value in the domain
Each value in the codomain can be the image of at most one value in the domain s t
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87. Example So we must ask if it is a function? Is it one-to-one? Well
Domain = {1, 2, 3,} Codomain = {1, 2, 3, 4} f(x) = x + 1
So we must ask if it is a function?
Choose 1: f(1) = 2
Choose 2: f(2) = 3
Choose 3: f(3) = 4
So it is a function.
Is it one-to-one? Well
we only reached the value 2 by using x = 1.
we only reached the value 3 by using x = 2.
we only reached the value 4 by using x = 3.
So it is one-to-one
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88. Is it a function? Is it one-to-one?
Domain = {-2, -1, 0, 1, 2} Codomain = {0, 1, 2, 3, 4, 5, 6} f(x) = x*x
Is it a function?
f(-2) = 4, f(-1) = 1, f(0) = 0, f(1) = 1, f(2)=4
So it is a function
Is it one-to-one?
No. We can reach the value 4 in two ways.
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89. Dealing with infinite sets
f: R->R with the rule f(x)=x2+4x+1
Is the function one-to-one ?
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90. Dealing with infinite sets (cont’d)
g: Q+->Q+ with the rule
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91. Dealing with muti-variate function
Is the following function one-to-one, onto, invertible ? 1. 2.
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92. Outline Definition of function
How to decide if a relation is a function or not
Property of functions
Onto
One-to-one
Pigeonhole principle
Function composition
Inverse function
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93. Function Composition
We can also put functions together – this is called composition
f ◦ g which reads “f compose g”
This means I insert the value of the function g into the function f.
f(x) = x+5, g(x) = 2x + 3
The composition says put the value of g into f
f ◦ g (x)=f(g(x))=f(2x+3) = (2x+3) + 5 = 2x + 8
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94. Assume we have two functions with Domains and Codomains over all integers
f(x) = 3x – 2
g(x) = x * x
What is f ◦ g ? What is g ◦ f? What is f ◦ f? What is g ◦ g ? What is f ◦ g for g(2)?
f(x*x) = 3(x*x)-2 = 3x2-2
g(3x-2) = (3x-2)*(3x-2)=9x2-12x+4
f(3x-2) = 3(3x-2)-2 = 9x-8
g(x*x) = (x*x) * (x*x) = x4
f(g(2)) = f(4) = 3(4) – 2 = 10
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95. Finding missing functions
If we know f(x), and (g ◦ f)(z), how to find out g(y) ?
ex. f(x)=3x-2, (g ◦ f)(z)=9z2-9z, what is g(y) ?
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96. Finding missing functions
If we know g(y), and (g◦ f)(z), how to find out f(x) ?
ex. g(y)=2y-1, (g◦ f)(z)=4z2-1, what is f(x) ?
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97. Bijection
A function that is both onto and one-to-one is called a bijection, or we say the function is bijective.
Illustration of functions and properties using figures
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98. Bijection and Inverse For any bijective function
Every value in the codomain has some value in domain mapped to it (because it’s onto)
Every value in the codomain has no more than one value mapped to it (because it’s one-to-one)
Therefore, every value in the codomain has one and only one value in the domain mapped to it
This means if we reverse the mapping, we get a function too.
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99. What is an inverse?
An inverse of a function is another function that reverses the process of the original function.
To create an inverse
Make the old Codomain the new domain
Make the old domain the new Codomain
Swap the f(x) and the x in the formula
Use algebra to get the f(x) back by itself
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100. Inverse Example New domain = {4, 8, 12, 16}
Domain = {2, 4, 6, 8} Codomain = {4, 8, 12, 16} f(x) = 2x
New domain = {4, 8, 12, 16}
New Codomain = {2, 4, 6, 8}
To compute the new formula reverse the f(x) and the x
x = 2f(x)
Then solve for f(x)
f(x) = x / 2 … so that is our inverse
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101. How to find inverse of a function ?
If the function is given by the set
If the function is given by a diagram
If the function is given by a formula
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