- 2+2 Club Piraeus
- Gopanel 2 2 0 2 Sezonas
- Gopanel 2 2 0 2 0 O N L I N E W O R K C O M
- 2.2 Gd
- 2+2 Online Tv
Return to the Lessons Index | Do the Lessons in Order | Print-friendly page |
Composition of Functions:
Composing with Sets of Points (page 1 of 6)
Composing with Sets of Points (page 1 of 6)
Sections: Composing functions that are sets of point, Composing functions at points, Composing functions with other functions, Word problems using composition, Inverse functions and composition
Until now, given a function f(x), you would plug a number or another variable in for x. You could even get fancy and plug in an entire expression for x. For example, given f(x) = 2x + 3, you could find f(y2 – 1) by plugging y2 – 1 in for x to get f(y2 – 1) = 2(y2 – 1) + 3 = 2y2 – 2 + 3 = 2y2 + 1.
In function composition, you're plugging entire functions in for the x. In other words, you're always getting 'fancy'. But let's start simple. Instead of dealing with functions as formulas, let's deal with functions as sets of (x, y) points:
- Let f = {(–2, 3), (–1, 1), (0, 0), (1, –1), (2, –3)} and
let g = {(–3, 1), (–1, –2), (0, 2), (2, 2), (3, 1)}.
Find (i)f (1), (ii) g(–1), and (iii) (gof )(1).
GoPanel 2 – Web Server Manager v2.2.0. AppStore QR-Code goPanel 2 - Web Server Manager. Developer: Global Web SRL. V2 comes with new features to the app, in short: – Nginx as Reverse Proxy on top of Apache – Redis (in-memory key-value store). Solve your math problems using our free math solver with step-by-step solutions. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more.
(i) This type of exercise is meant to emphasize that the (x, y) points are really (x, f (x)) points. To find f (1), I need to find the (x, y) point in the set of (x, f (x)) points that has a first coordinate of x = 1. Then f (1) is the y-value of that point. In this case, the point with x = 1 is (1, –1), so:
Advertisement
f (1) = –1
(ii) The point in the g(x) set of point with x = –1 is the point (–1, –2), so:
g(–1) = –2
GoPanel 1.2.0 – Manage Web servers. June 27, 2016 goPanel is an incredibly intuitive OS X app for the management of web servers, an alternative to existing control-panel apps you install on Unix-based servers for web hosting. GoPanel 1.2.0 – Manage Web servers. June 27, 2016 goPanel is an incredibly intuitive OS X app for the management of web servers, an alternative to existing control-panel apps you install on Unix-based servers for web hosting. GoPanel is an incredibly intuitive OS X app for the management of web servers, an alternative to existing control-panel apps you install on Unix-based servers for web hosting. Easy-to-install and configure Apache or Nginx webserver, PHP, MySQL, FTP, domains, free SSL certs and emails on your server. GoPanel lets you easily connect and manage.
(iii) What is '(gof )(1)'? This is read as 'g-compose-f of 1', and means 'plug 1 into f, evaluate, and then plug the result into g'. The computation can feel a lot easier if I use the following, more intuitive, formatting:
(gof )(1) = g( f(1))
Now I'll work in steps, keeping in mind that, while I may be used to doing things from the left to the right (because that's how we read), composition works from the right to the left (or, if you prefer, from the inside out). So I'll start with the x = 1. I am plugging this into f(x), so I look in the set of f(x) points for a point with x = 1. The point is (1, –1). This tells me that f(1) = –1, so now I have: Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved
(gof )(1) = g( f(1)) = g(–1)
Working from the right back toward the left, I am now plugging x = –1 (from 'f(1) = –1') into g(x), so I look in the set of g(x) points for a point with x = –1. That point is (–1, –2). This tells me that g(–1) = –2, so now I have my answer:
(gof )(1) = g( f(1)) = g(–1) = –2
Note that they never told us what were the formulas, if any, for f(x) or g(x); we were only given a list of points. But this list was sufficient for answering the question, as long as we keep track of our x- and y-values.
- Let f = {(–2, 3), (–1, 1), (0, 0), (1, –1), (2, –3)} and
let g = {(–3, 1), (–1, –2), (0, 2), (2, 2), (3, 1)}.
Find (i) ( fog)(0), (ii)( fog)(–1), and (iii)(gof )(–1).
(i) To find ( fog)(0), ('f-compose-g of zero'), I'll rewrite the expression as:
Jixipix premium pack 1 6 1. ( fog)(0) = f(g(0))
This tells me that I'm going to plug zero into g(x), simplify, and then plug the result into f(x). Looking at the list of g(x) points, I find (0, 2), so g(0) = 2, and I need now to find f(2). Looking at the list of f(x) points, I find (2, –3), so f(2) = –3. Then:
( fog)(0) = f(g(0)) = f(2) = –3
(ii) The second part works the same way:
( fog)(–1) = f(g(–1)) = f(–2) = 3
(iii) I can rewrite the composition as (gof )(–1) = g( f(–1)) = g(1).
Uh-oh; there is no g(x) point with x = 1, so it is nonsense to try to find the value of g(1). In math-speak, g(1) is 'not defined'; that is, it is nonsense.Then (gof )(–1) is also nonsense, so the answer is:
(gof )(–1) is undefined.
Part (iii) of the above example points out an important consideration regarding domains and ranges. It may be that your composed function (the result you get after composing two other functions) will have a restricted domain, or at least a domain that is more restricted than you might otherwise have expected. This will be more important when we deal with composing functions symbolically later.
Another exercise of this type gives you two graphs, rather than two sets of points, and has you read the points (the function values) from these graphs.
- Given f(x) and g(x) as shown below, find ( fog)(–1).
In this case, I will read the points from the graph. I've been asked to find ( fog)(–1) = f(g(–1)). This means that I first need to find g(–1). So I look on the graph of g(x), and find x = –1. Tracing up from x = –1 to the graph of g(x), I arrive at y = 3. Then the point (–1, 3) is on the graph of g(x), and g(–1) = 3.
Now I plug this value, x = 3, into f(x). To do this, I look at the graph of f(x) and find x = 3. Tracing up from x = 3 to the graph of f(x), I arrive at y = 3. Then the point (3, 3) is on the graph of f(x), and f(3) = 3.
Then( fog)(–1) = f(g(–1)) = f(3) = 3.
- Given f(x) and g(x) as shown in the graphs below, find (gof )(x) for integral values of x on the interval –3 <x< 3.
f(x): | g(x): |
This is asking me for all the values of (gof )(x) = g( f(x)) for x = –3, –2, –1, 0, 1, 2, and 3. So I'll just follow the points on the graphs and compute all the values:
(gof )(–3) = g( f(–3)) = g(1) = –1
I got this answer by looking at x = –3 on the f(x) graph, finding the corresponding y-value of 1 on the f(x) graph, and using this answer as my new x-value on the g(x) graph. That is, I looked at x = –3 on the f(x) graph, found that this led to y = 1, went to x = 1 on the g(x) graph, and found that this led to y = –1. Similarly:
(gof )(–2) = g( f(–2)) = g(–1) = 3
(gof )(–1) = g( f(–1)) = g(–3) = –2
(gof )(0) = g( f(0)) = g(–2) = 0
(gof )(1) = g( f(1)) = g(0) = 2
(gof )(2) = g( f(2)) = g(2) = –3
(gof )(3) = g( f(3)) = g(3) = 1
(gof )(–1) = g( f(–1)) = g(–3) = –2
(gof )(0) = g( f(0)) = g(–2) = 0
(gof )(1) = g( f(1)) = g(0) = 2
(gof )(2) = g( f(2)) = g(2) = –3
(gof )(3) = g( f(3)) = g(3) = 1
You aren't generally given functions as sets of points or as graphs, however. Generally, you have formulas for your functions. So let's see what composition looks like in that case..
2+2 Club Piraeus
Top | 1 | 2 | 3 | 4 | 5|6| Return to IndexNext >>
Cite this article as: | Stapel, Elizabeth. 'Composing with Sets of Points.' Purplemath. Available from https://www.purplemath.com/modules/fcncomp.htm. Accessed [Date] [Month] 2016 |
Developer: Global Web SRL
goPanel 2 offers a reliable way to manage Linux Web Servers directly from your Mac’s Desktop / Laptop, an alternative to existing control panel software you install on your Unix based servers for web hosting.
v2 comes with new features to the app, in short:
– Nginx as Reverse Proxy on top of Apache
– Redis (in-memory key-value store)
– Varnish (HTTP accelerator designed for content-heavy dynamic web sites)
– Removed FTP as mandatory dependency for the webserver
– Added Custom Scripts tab in Config so we allow any user to contribute with own scripts for a distro and start creating a community
– Nginx as Reverse Proxy on top of Apache
– Redis (in-memory key-value store)
– Varnish (HTTP accelerator designed for content-heavy dynamic web sites)
– Removed FTP as mandatory dependency for the webserver
– Added Custom Scripts tab in Config so we allow any user to contribute with own scripts for a distro and start creating a community
***** This is the best Linux Web Server manager on App Store *****
Gopanel 2 2 0 2 Sezonas
Easy to install and configure Apache or Nginx web server, PHP, MySQL or MariaDB, FTP, SSL certificates, domains and emails on your server.
Incredible Features:
– Add&manage UNLIMITED servers (VPS or Dedicated)
– Install, configure and manage: Apache or Nginx, PHP, FTP (Pure-FTPd), MySQL or MariaDB, Mail Server to get each of your servers ready to host domains
– PHP & Apache on/off from selection of modules
– Unlimited MySQL/MariaDB users and databases, domains, ftp accounts and emails
– Unlimited FREE SSL** certs issued by Let’s Encrypt certificate authority
– Shell account for FTP users
– Fail2Ban intrusion prevention software Install, Configure and Latest Activity
– Setup scheduled cron jobs
– Setup backup for your files or databases locally or on Amazon S3 and FTP external backup
– View server logs and block IP’s
– Rollback up to 50 earlier versions of your config files in case you need to
– System updates – keep your linux server up to date
– 3rd Party Scripts
— WP-CLI (command-line tools for managing WordPress installations)
— ConfigServer Security & Firewall (csf)
— Composer (application-level package manager)
— PHPMyAdmin(database manager)
— Webmailer (roundcube)
— OneClick WordPress Installer
– Add&manage UNLIMITED servers (VPS or Dedicated)
– Install, configure and manage: Apache or Nginx, PHP, FTP (Pure-FTPd), MySQL or MariaDB, Mail Server to get each of your servers ready to host domains
– PHP & Apache on/off from selection of modules
– Unlimited MySQL/MariaDB users and databases, domains, ftp accounts and emails
– Unlimited FREE SSL** certs issued by Let’s Encrypt certificate authority
– Shell account for FTP users
– Fail2Ban intrusion prevention software Install, Configure and Latest Activity
– Setup scheduled cron jobs
– Setup backup for your files or databases locally or on Amazon S3 and FTP external backup
– View server logs and block IP’s
– Rollback up to 50 earlier versions of your config files in case you need to
– System updates – keep your linux server up to date
– 3rd Party Scripts
— WP-CLI (command-line tools for managing WordPress installations)
— ConfigServer Security & Firewall (csf)
— Composer (application-level package manager)
— PHPMyAdmin(database manager)
— Webmailer (roundcube)
— OneClick WordPress Installer
Linux distributions supported:
– Ubuntu 14.04 (LTS), 14.10, 15.04, 15.10, 16.04
– CentOS 6.x, 7.x
– Amazon Linux 2016.03
– Red Hat Enterprise Linux 6.x, 7.x
– Debian 7 and 8
and YES goPanel works perfect with Amazon instances and Digital Ocean droplets as long as you use a linux distribution we support.
– Ubuntu 14.04 (LTS), 14.10, 15.04, 15.10, 16.04
– CentOS 6.x, 7.x
– Amazon Linux 2016.03
– Red Hat Enterprise Linux 6.x, 7.x
– Debian 7 and 8
and YES goPanel works perfect with Amazon instances and Digital Ocean droplets as long as you use a linux distribution we support.
* make sure your linux server does not have FTP server installed and you install it from the goPanel app.
** Let’s Encrypt – centOS 6.x experimental support only
** Let’s Encrypt – AMI linux not supported yet
** Let’s Encrypt – centOS 6.x experimental support only
** Let’s Encrypt – AMI linux not supported yet
What’s new in goPanel
Information
Gopanel 2 2 0 2 0 O N L I N E W O R K C O M
Size 11 MB
Compatibility OS X 10.10 or later, 64-bit processor
Age Rating Rated 4+
2.2 Gd
Price $38.99