<< Chapter < Page
  Functions   Page 1 / 1
Chapter >> Page >
This module defines the domain and range of a function.

Consider the function y = x size 12{y= sqrt {x} } {} . If this function is given a 9 it hands back a 3. If this function is given a 2 it hands back…well, it hands back 2 size 12{ sqrt {2} } {} , which is approximately 1.4. The answer cannot be specified exactly as a fraction or decimal, but it is a perfectly good answer nonetheless.

On the other hand, what if this function is handed –4? There is no 4 size 12{ sqrt { - 4} } {} , so the function has no number to hand back. If our function is a computer or calculator, it responds with an error message. So we see that this function is able to respond to the numbers 9 and 2, but it is not able to respond in any way to the number -4. Mathematically, we express this by saying that 9 and 2 are in the “domain” of the square root function, and –4 is not in the domain of this function.

Domain
The domain of a function is all the numbers that it can successfully act on. Put another way, it is all the numbers that can go into the function.

A square root cannot successfully act on a negative number. We say that “The domain of x size 12{ sqrt {x} } {} is all numbers such that ” meaning that if you give this function zero or a positive number, it can act on it; if you give this function a negative number, it cannot.

A subtler example is the function y = x + 7 size 12{y= sqrt {x+7} } {} . Does this function have the same domain as the previous function? No, it does not. If you hand this function a –4 it successfully hands back 3 size 12{ sqrt {3} } {} (about 1.7). –4 is in the domain of this function. On the other hand, if you hand this function a –8 it attempts to take 1 size 12{ sqrt { - 1} } {} and fails; –8 is not in the domain of this function. If you play with a few more numbers, you should be able to convince yourself that the domain of this function is all numbers x such that x 7 size 12{x>= - 7} {} .

You are probably familiar with two mathematical operations that are not allowed . The first is, you are not allowed to take the square root of a negative number. As we have seen, this leads to restrictions on the domain of any function that includes square roots.

The second restriction is, you are not allowed to divide by zero. This can also restrict the domain of functions. For instance, the function y = 1 x 2 4 size 12{y= { {1} over {x rSup { size 8{2} } - 4} } } {} has as its domain all numbers except x = 2 size 12{x=2} {} and x = 2 size 12{x= - 2} {} . These two numbers both cause the function to attempt to divide by 0, and hence fail. If you ask a calculator to plug x = 2 size 12{x=2} {} into this function, you will get an error message.

So: if you are given a function, how can you find its domain? Look for any number that puts a negative number under the square root; these numbers are not in the domain. Look for any number that causes the function to divide by zero; these numbers are not in the domain. All other numbers are in the domain.

Function Domain Comments
x size 12{ sqrt {x} } {} x 0 size 12{x>= 0} {} You can take the square root of 0, or of any positive number, but you cannot take the square root of a negative number.
x + 7 size 12{ sqrt {x+7} } {} x 7 size 12{x>= - 7} {} If you plug in any number greater than or equal to –7, you will be taking a legal square root. If you plug in a number less than –7 , you will be taking the square root of a negative number.This domain can also be understood graphically: the graph y = x size 12{y= sqrt {x} } {} has been moved 7 units to the left. See “horizontal permutations” below.
1 x size 12{ { {1} over {x} } } {} x 0 size 12{x<>0} {} In other words, the domain is “all numbers except 0.” You are not allowed to divide by 0. You are allowed to divide by anything else.
1 x 3 size 12{ { {1} over {x - 3} } } {} x 3 size 12{x<>3} {} If x = 3 size 12{x=3} {} then you are dividing by 0, which is not allowed. If x = 0 size 12{x=0} {} you are dividing by –3, which is allowed. So be careful! The rule is not “when you are dividing, x cannot be 0.” The rule is “ x size 12{x} {} can never be any value that would put a 0 in the denominator.”
1 x 2 4 size 12{ { {1} over {x rSup { size 8{2} } - 4} } } {} x ± 2 size 12{x<>+- 2} {} Or, “ x size 12{x} {} can be any number except 2 or –2.” Either of these x size 12{x} {} values will put a 0 in the denominator, so neither one is allowed.
2 x + x 2 3x + 4 size 12{2 rSup { size 8{x} } +x rSup { size 8{2} } - 3x+4} {} All numbers You can plug any x size 12{x} {} value into this function and it will come back with a number.
x 3 x 5 size 12{ { { sqrt {x - 3} } over {x - 5} } } {} x 3 x 5 alignl { stack { size 12{x>= 3} {} # size 12{x<>5} {} } } {} In words, the domain is all numbers greater than or equal to 3, except the number 5 . Numbers less than 3 put negative numbers under the square root; 5 causes a division by 0.

You can confirm all these results with your calculator; try plugging numbers into these functions, and see when you get errors!

A related concept is range .

Range
The range of a function is all the numbers that it may possibly produce. Put another way, it is all the numbers that can come out of the function.

To illustrate this example, let us return to the function y = x + 7 size 12{y= sqrt {x+7} } {} . Recall that we said the domain of this function was all numbers x size 12{x} {} such that x 7 size 12{x>= - 7} {} ; in other words, you are allowed to put any number greater than or equal to –7 into this function.

What numbers might come out of this function? If you put in a –7 you get out a 0. ( 0 = 0 size 12{ sqrt {0} =0} {} ) If you put in a –6 you get out 1 = 1 size 12{ sqrt {1} =1} {} . As you increase the x value, the y values also increase. However, if you put in x = 8 size 12{x= - 8} {} nothing comes out at all. Hence, the range of this function is all numbers y size 12{y} {} such that y 0 size 12{y>= 0} {} . That is, this function is capable of handing back 0 or any positive number, but it will never hand back a negative number.

It’s easy to get the words domain and range confused—and it’s important to keep them distinct, because although they are related concepts, they are different from each other. One trick that sometimes helps is to remember that, in everyday useage, “your domain” is your home, your land—it is where you begin. A function begins in its own domain. It ends up somewhere out on the range.

A different notation for domain and range

Domains and ranges above are sometimes expressed as intervals, using the following rules:

  • Parentheses ( ) mean “an interval starting or ending here, but not including this number”
  • Square brackets [ ] mean “an interval starting or ending here, including this number”

This is easiest to explain with examples.

This notation... ...means this... ...or in other words
( 3,5 ) size 12{ \( - 3,5 \) } {} All numbers between –3 and 5, not including –3 and 5. 3 < x < 5 size 12{ - 3<x<5} {}
[ 3,5 ] size 12{ \[ - 3,5 \] } {} All numbers between –3 and 5, including –3 and 5. 3 x 5 size 12{ - 3<= x<= 5} {}
[ 3,5 ) size 12{ \[ - 3,5 \) } {} All numbers between –3 and 5, including –3 but not 5. 3 x < 5 size 12{ - 3<= x<5} {}
( , 10 ] size 12{ \( - infinity ,"10" \] } {} All numbers less than or equal to 10. x 10 size 12{x<= "10"} {}
( 23 , ) size 12{ \( "23", infinity \) } {} All numbers greater than 23. x > 23 size 12{x>"23"} {}
( , 4 ) size 12{ \( - infinity ,4 \) } {} ( 4, ) size 12{ \( 4, infinity \) } {} All numbers less than 4, and all numbers greater than 4. In other words, all numbers except 4. x 4 size 12{x<>4} {}

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Functions. OpenStax CNX. Feb 01, 2011 Download for free at http://cnx.org/content/col11272/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Functions' conversation and receive update notifications?

Ask