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## #1 2009-09-13 15:52:41

MathsIsFun
Registered: 2005-01-21
Posts: 7,630

Just fnishing off some new pages on functions:

Evaluating Functions

Odd and Even Functions

Increasing Functions

Comments, etc welcome. Help make the pages perfect

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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## #2 2009-09-14 08:20:04

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 106,337

Hi MathsisFun;

Look good from here, thanks.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
No great discovery was ever made without a bold guess.

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## #3 2009-09-14 10:19:14

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

I learnt increasing functions as x < y  ⇒ f(x) ≤ f(y).
ie. A constant function counts as increasing, for example.

I learnt the definition on the page as "strictly increasing".

Probably different people use different definitions though.

Why did the vector cross the road?
It wanted to be normal.

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## #4 2009-09-14 10:26:00

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 106,337

Hi;

I just read monotonic increasing. Looked it up on wikipedia and they said the same thing for x < y  ⇒ f(x) ≤ f(y). Preserving the order they call it.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
No great discovery was ever made without a bold guess.

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## #5 2009-09-14 13:40:24

MathsIsFun
Registered: 2005-01-21
Posts: 7,630

Good points, thank you.

I have redone Increasing Functions

Please check it out and let me know if it passes.

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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## #6 2009-09-14 16:55:24

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 106,337

Hi MathsIsFun;

Glad you went with strictly increasing instead of monotonic. Thanks for the page.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
No great discovery was ever made without a bold guess.

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## #7 2009-09-14 23:13:30

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

According to the new definition, a constant function is increasing and decreasing, rather than neither.

Also, I'd say something about how for local extrema, the point should be "in" the interval, rather than on the edge.

For example, with f(x) = x on the interval [1,2], the maximum there would be at x=2. But you wouldn't consider that a local maximum of the function.

Why did the vector cross the road?
It wanted to be normal.

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## #8 2009-09-14 23:46:17

MathsIsFun
Registered: 2005-01-21
Posts: 7,630

mathsyperson wrote:

According to the new definition, a constant function is increasing and decreasing, rather than neither.

That seems to be a consequence of the definitions, yes. But not strictly increasing or strictly decreasing.

mathsyperson wrote:

Also, I'd say something about how for local extrema, the point should be "in" the interval, rather than on the edge.

For example, with f(x) = x on the interval [1,2], the maximum there would be at x=2. But you wouldn't consider that a local maximum of the function.

In that case there would be NO local extrema?

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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## #9 2009-09-15 06:42:03

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Well, that problem only happens when you work with a specific interval. So maybe the page should keep the "f(a) ≥ f(x) for all x in the interval", but say it has to be true for all small enough intervals rather than just one.

Or is that complicating things too much?

Why did the vector cross the road?
It wanted to be normal.

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## #10 2009-09-15 13:13:13

MathsIsFun
Registered: 2005-01-21
Posts: 7,630

I added "f(a) should be inside the interval, not at one end or the other."

Hopefully that covers it

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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## #11 2009-09-15 13:28:57

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

A Constant Function neither increases nor decreases:

You realize that this is wrong? In fact, a Constant Function is both increasing and decreasing (but not strictly) along its whole domain.

Last edited by JaneFairfax (2009-09-15 13:29:23)

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## #12 2009-09-15 13:35:06

MathsIsFun
Registered: 2005-01-21
Posts: 7,630

Missed that one, thanks.

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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## #13 2009-09-15 13:41:31

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

According to the definition, a function y=f(x) is increasing iff

when x1 < x2 then f(x1) ≤ f(x2)

Hence, if y=f(x) is non-increasing, then there exist x1 < x2 such that f(x1) > f(x2). But a constant function cannot possibly have f(x1) > f(x2). Therefore a constant function cannot be non-increasing.

By a similar argument a constant function cannot be non-decreasing either. Therefore it must be both increasing and decreasing.

Last edited by JaneFairfax (2009-09-15 17:23:45)

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## #14 2009-09-15 14:07:13

MathsIsFun
Registered: 2005-01-21
Posts: 7,630

I agree.

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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## #15 2009-09-15 19:05:59

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868