# S.O.S. Mathematics CyberBoard

Your Resource for mathematics help on the web!
 It is currently Sun, 19 May 2013 08:12:48 UTC

 All times are UTC [ DST ]

 Page 1 of 1 [ 10 posts ]
 Print view Previous topic | Next topic
Author Message
 Post subject: even and decreasingPosted: Thu, 10 May 2012 17:46:15 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Fri, 28 Dec 2007 12:01:53 UTC
Posts: 1261
Is it possible that function is even as well as decreasing in its domain?

_________________
There is no god in this world except PARENTS and i have lost ONE

Top

 Post subject: Re: even and decreasingPosted: Thu, 10 May 2012 19:25:58 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12075
Location: Austin, TX
mun wrote:
Is it possible that function is even as well as decreasing in its domain?

Of course, let the domain be and let the function be .

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination

Top

 Post subject: Re: even and decreasingPosted: Thu, 10 May 2012 23:44:42 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Fri, 1 Jul 2011 01:17:26 UTC
Posts: 321
mun wrote:
Is it possible that function is even as well as decreasing in its domain?

No. If the function is decreasing for x > 0, it must be increasing for x < 0 and vice versa.

y = -x^2 is a parabola pointing downward, so it is decreasing for X > 0 and increasing for x < 0, with the max at x = 0.

Top

 Post subject: Re: even and decreasingPosted: Fri, 11 May 2012 00:06:38 UTC
 S.O.S. Oldtimer

Joined: Fri, 27 Jul 2007 10:17:26 UTC
Posts: 278
Location: Chandler, AZ, USA
mun wrote:
Is it possible that function is even as well as decreasing in its domain?

Of course, let the domain be and let the function be .

Since the definition of an even function is that , can you have an even function whose domain is on only one side of zero?

Top

 Post subject: Re: even and decreasingPosted: Fri, 11 May 2012 00:22:28 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12075
Location: Austin, TX
mathematic wrote:
mun wrote:
Is it possible that function is even as well as decreasing in its domain?

No. If the function is decreasing for x > 0, it must be increasing for x < 0 and vice versa.

y = -x^2 is a parabola pointing downward, so it is decreasing for X > 0 and increasing for x < 0, with the max at x = 0.

My function is only defined on you said "on its domain", and in its domain it is indeed decreasing. In fact, I can take any decreasing function on and with that domain restriction we can say it satisfies the condition vacuously.

If you demand that the function be defined everywhere, or on a neighborhood of 0, then the answer should be clear to you as a "no" by the MVT (unless you allow for weakly decreasing, in which case 0 works), but as-stated my example works just fine.

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination

Top

 Post subject: Re: even and decreasingPosted: Fri, 11 May 2012 00:23:55 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12075
Location: Austin, TX
alstat wrote:
mun wrote:
Is it possible that function is even as well as decreasing in its domain?

Of course, let the domain be and let the function be .

Since the definition of an even function is that , can you have an even function whose domain is on only one side of zero?

The definition is such that , it just so happens that my set of such that is also in the domain is empty, so since we have a universal quantifier applied to an empty set, it is vacuously true that my function is even.

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination

Top

 Post subject: Re: even and decreasingPosted: Sat, 12 May 2012 18:12:56 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Fri, 28 Dec 2007 12:01:53 UTC
Posts: 1261
alstat wrote:
mun wrote:
Is it possible that function is even as well as decreasing in its domain?

Of course, let the domain be and let the function be .

Since the definition of an even function is that , can you have an even function whose domain is on only one side of zero?

The definition is such that , it just so happens that my set of such that is also in the domain is empty, so since we have a universal quantifier applied to an empty set, it is vacuously true that my function is even.

Good evening sir
sorry sir but one of my great sir has given this explanation (which confused me more )as he disagree with your explanation .His explanation is following
Any of the common definitions for even functions
A function such that for all in the domain of ; or

A function having its graph symmetrical with respect to the axis;

in fact implies that the domain of an even function is itself symmetrical with respect to the origin, i.e. . This is for example how the notion of even function is introduced in Romanian textbooks: a function where with , and for all .
Thus speaking about an even function defined on is disallowed, and not vacuously true (only because does not exist).

_________________
There is no god in this world except PARENTS and i have lost ONE

Top

 Post subject: Re: even and decreasingPosted: Sat, 12 May 2012 18:22:01 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12075
Location: Austin, TX
mun wrote:
alstat wrote:
mun wrote:
Is it possible that function is even as well as decreasing in its domain?

Of course, let the domain be and let the function be .

Since the definition of an even function is that , can you have an even function whose domain is on only one side of zero?

The definition is such that , it just so happens that my set of such that is also in the domain is empty, so since we have a universal quantifier applied to an empty set, it is vacuously true that my function is even.

Good evening sir
sorry sir but one of my great sir has given this explanation (which confused me more )as he disagree with your explanation .His explanation is following
Any of the common definitions for even functions
A function such that for all in the domain of ; or

A function having its graph symmetrical with respect to the axis;

in fact implies that the domain of an even function is itself symmetrical with respect to the origin, i.e. . This is for example how the notion of even function is introduced in Romanian textbooks: a function where with , and for all .
Thus speaking about an even function defined on is disallowed, and not vacuously true (only because does not exist).

I've never seen that assumption made, before. If that is your definition of an even function, then you may ignore what I've said, but I don't see any reason why one should assume the domain is symmetric and I never have. It's easy to *extend* even functions to a maximal, symmetric domain, but there's no reason do do that ahead of time IMO.

In that case you need to see if weakly decreasing is possible, if so, then the identically zero function works, otherwise the answer is no for the reason explained above (I'm assuming the function is because this is a calculus question, you can get it for continuous functions without too much work as well.

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination

Top

 Post subject: Re: even and decreasingPosted: Sat, 12 May 2012 18:27:14 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6004
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
mun wrote:
Is it possible that function is even as well as decreasing in its domain?

Of course, let the domain be and let the function be .

On the other hand, letting gives you vacuously f is decreasing, whatever ordered set happens to be mapping to.

_________________

Top

 Post subject: Re: even and decreasingPosted: Sat, 12 May 2012 18:37:36 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12075
Location: Austin, TX
outermeasure wrote:
mun wrote:
Is it possible that function is even as well as decreasing in its domain?

Of course, let the domain be and let the function be .

On the other hand, letting gives you vacuously f is decreasing, whatever ordered set happens to be mapping to.

and a nice countable set would be workable as well. I love the magic of calculus.

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination

Top

 Display posts from previous: All posts1 day7 days2 weeks1 month3 months6 months1 year Sort by AuthorPost timeSubject AscendingDescending
 Page 1 of 1 [ 10 posts ]

 All times are UTC [ DST ]

#### Who is online

Users browsing this forum: No registered users

 You cannot post new topics in this forumYou cannot reply to topics in this forumYou cannot edit your posts in this forumYou cannot delete your posts in this forum

Search for:
 Jump to:  Select a forum ------------------ High School and College Mathematics    Algebra    Geometry and Trigonometry    Calculus    Matrix Algebra    Differential Equations    Probability and Statistics    Proposed Problems Applications    Physics, Chemistry, Engineering, etc.    Computer Science    Math for Business and Economics Advanced Mathematics    Foundations    Algebra and Number Theory    Analysis and Topology    Applied Mathematics    Other Topics in Advanced Mathematics Other Topics    Administrator Announcements    Comments and Suggestions for S.O.S. Math    Posting Math Formulas with LaTeX    Miscellaneous