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#1 2007-06-22 13:21:34
- MathsIsFun
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What is a Function?
I have been a bit adventurous and written a draft page on What is a Function?.
I start off by explaining that the function is like a machine, but that means I then had to explain that it's not really a machine, because it doesn't destroy the import, and it happens all a once. Can anyone think of a better strategy?
Also... in my attempts to make it simple, may also have got it wrong. So if you like to read it with a critical eye, that would be appreciated.
Continuing the adventure, I also wrote drafts for Domain, Range and Codomain and Injective, Surjective and Bijective.
The same thing applies, please read with a critical eye, and let me know what you find, good or bad.
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
#2 2007-06-22 13:56:52
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Re: What is a Function?
Is f(x) = √x a function?
Generally speaking ... NO! ... is that a surpise?
The reason is that there could be two answers, for example f(9) = √9 = 3 or -3
This is not correct, by normal standards. Square root is a function, this is why when solving equations we typically write something such as:
We need the plus or minus in there because when we say "square root" we mean the positive root. It would be nice to mention that we define square root this way because we want it to be a function.
If you want an example of something that isn't a function, sin-¹ is not. It only becomes a function when the domain of sin is restricted to (-pi, pi). A picture with the vertical line test would help as not many know what sin-¹ looks like.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
#3 2007-06-22 16:12:49
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Re: What is a Function?
So if there is no
infront of the square root, it means it is positive?#4 2007-06-22 16:20:11
- Ricky
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Re: What is a Function?
Correct.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
#5 2007-06-22 17:39:19
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Re: What is a Function?
Ricky wrote:
A picture with the vertical line test would help as not many know what sin-¹ looks like.
Good idea. Plot looks like this: Plot of asin(x).
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
#6 2007-06-22 18:22:31
- JaneFairfax
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Re: What is a Function?
Is f(x) = √x a function?
Generally speaking ... NO! ... is that a surpise?
The reason is that there could be two answers, for example f(9) = √9 = 3 or -3This is not correct, by normal standards. Square root is a function …
Square root is indeed a function for non-negative real numbers. However, it is not a function for negative real numbers or complex numbers (since, for complex numbers in general, there is no positive or negative). For example, √(−1) = ±i.
MathIsFun, you might like to add a little advanced technical stuff about the formal definition of a function: this is the usual set-theoretic definition of a function: 
Last edited by JaneFairfax (2007-06-22 18:36:16)
#7 2007-06-23 00:33:00
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Re: What is a Function?
Ah, true Jane. In that case, I would stick with square root and leave out sin-¹, and say that the square root is a function over positive real numbers, but it isn't a function over all real numbers, using -1 as an example. I would also make sure to emphasize with this example that whether something is a function or not can depend on what set you use for the domain, and that you can use just about any set you want.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
#8 2007-06-25 11:33:47
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Re: What is a Function?
Redid What is a Function? and Domain, Range and Codomain.
I use "square root" and contrast it with √, plus I added the vertical line test and a conclusion.
Better? Any errors?
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
#9 2007-06-27 18:33:23
- justlookingforthemoment
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Re: What is a Function?
I can't seem to see the 'not single valued' image. It just shows as text.
#10 2007-06-27 19:00:00
- JaneFairfax
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Re: What is a Function?
The image with the alt text “not single valued" seems to be missing. 
#11 2007-06-27 20:15:19
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Re: What is a Function?
Ahh, thanks, fixed it.
Is it all good, then? Shall I link to them from my "Recent Additions" page now?
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
#12 2007-07-01 16:10:05
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Re: What is a Function?
Yes. Now that the problem is fixed, it may be added.
Character is who you are when no one is looking.
#13 2007-07-01 16:33:16
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Re: What is a Function?
Thanks g-man!
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
#14 2009-09-14 13:55:43
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Re: What is a Function?
I have redone the What is a Function? page.
Comments and suggestions welcome.
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
#15 2009-09-14 17:08:10
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Re: What is a Function?
Hi MathsisFun;
It is a little thing but the top the page says sqrt(x) is a function (examples of functions). While at the bottom there is a graph (vertical line test) showing that multivalued outputs (sqrt is multivalued) are not functions. Otherwise it is excellent and very functional. I kept this page for reference.
Last edited by bobbym (2009-09-14 17:20:25)
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#16 2009-09-14 17:49:26
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Re: What is a Function?
Thanks! Will fix.
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
#17 2009-10-08 15:41:05
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Re: What is a Function?
Well you wouldn't believe it, but I have been working on this page again. I wanted to make it clear that a function is a special type of relationship (a relationship may be valid but not a function).
It would be nice for you guys to read it from top to bottom to make sure I have it all correct ... and that it reads nicely. This is an important page for people to understand if they want to do well later on.
(Link again, for reference)
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
#18 2009-10-08 18:54:57
- bobbym
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Re: What is a Function?
Hi MathsisFun;
This is an important page for people to understand if they want to do well later on.
I agree, your fundamentals determine your success.
Great work!
There is a mistake in spelling 1/2 way down the page: folloows -> follows, just a typing error.
The definition on the page about explicit vs implicit maybe could be more rigorous.
Def: An implicit function is a function in which the dependent variable has not been given "explicitly" in terms of the independent variable.
Is the one I like but maybe yours is better. Or maybe I didn't fully grasp yours. I don't know.
Anyway, liked the page. Thanks for the work.
Last edited by bobbym (2009-10-08 19:00:22)
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.