Math Is Fun Forum

ZHero

Real Member

Registered: 2008-06-08

Posts: 1,889

Even or Odd or Constant function?

Is

an Even or Odd or Constant function?
Plz Justify...

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If two or more thoughts intersect, there has to be a point!

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Even or Odd or Constant function?

All of them.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Even or Odd or Constant function?

Hi;

So it's odd.

So it's even.

So it is constant.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

ZHero

Real Member

Registered: 2008-06-08

Posts: 1,889

Re: Even or Odd or Constant function?

mmm...

my Doubt here is...
can i write f(x)=0??
can i Simplify the Terms in the Definition of the function?

somebody told me that i CAN NOT do this...

Last edited by ZHero (2010-03-24 02:30:27)

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If two or more thoughts intersect, there has to be a point!

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Even or Odd or Constant function?

mmm...

my Doubt here is...
can i write f(x)=0??
can i Simplify the Terms in the Definition of the function?

somebody told me that i CAN NOT do this...

Of course you can write ƒ(x) = 0. But you also have to specify that the domain is

and not

. Whoever said you can’t probably meant that you can’t just write

and leave it there, but also have to indicate that the function is not defined at 0.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Even or Odd or Constant function?

Hi Jane;

This is how I understand it.

Some functions have a removable singularity.

Def: When the function is bounded in a neighborhood around a singularity ( Uh oh, neighborhoods, topolgy talk, I'm in trouble),  the function can be redefined at the point to remove it; hence it is known as a removable singularity. In contrast, when a function tends to infinity as z approaches 0; thus, it is not bounded and the singularity is not removable,

According to this and I found other sites that agree. Just google for removable singularity.

f(x) = sin(x) / x  , f(0) is defined and is equal to 1 This is the example they all give for a removable singularity.

f(x) above is defined at 0 because it is a removable singularity and I think the simplification is allowed. Or where am I going wrong?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Even or Odd or Constant function?

No, ƒ is not defined at 0.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Even or Odd or Constant function?

Hi Jane;

Def: When the function is bounded in a neighborhood around a singularity ( Uh oh, neighborhoods, topolgy talk, I'm in trouble),  the function can be redefined at the point to remove it; hence it is known as a removable singularity. In contrast, when a function tends to infinity as z approaches 0; thus, it is not bounded and the singularity is not removable,

According to this and I found other sites that agree. Just google for removable singularity.

f(x) = sin(x) / x  , f(0) is defined and is equal to 1 This is the example they all give for a removable singularity.

Why does the above definition apply to f(x) =sin(x) / x for f(0) and not for

at f(0). The limit exists from both sides and equals 0. Especially since f(x) is no different than 0 when it is simplified.

As you can see I am not understanding it.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Even or Odd or Constant function?

Hi Jane;

This exchange took place between me and another person yesterday.

Definition: When the function is bounded in a neighborhood around a singularity,the function can be redefined at the point to remove it;
hence it is known as a removable singularity. In contrast, when a
function tends to infinity as z approaches 0; thus, it is not bounded
and the singularity is not removable,
I found other sites that agree when I google for removable singularity.

The term "removable singularity", and in fact the term "singularity"
itself, is used primarily in complex analysis. Is that the context
of your question, or are the examples below intended to be functions
of a real variable? I ask because the definition you give above does
not seem to work for real functions. Also, you mention approaching
zero "from both sides", which would seem to imply that x lies on a
real number line rather than in a complex plane. I too use the
term "removable singularity" in real contexts because I don't know
an alternative; but we can't freely import every fact about them
from complex to real analysis.

>f(x) = sin(x) / x  , f(0) is defined and is equal to 1 This is the
>example they all give for a removable singularity.

Are you sure this is exactly what they say? I would say rather that f
(0) CAN be defined TO BE EQUAL to 1. Let me restate this in greater
detail. The function

f(x) = sin(x)/x

is NOT defined for x=0. However, we can define a NEW function

g(x) = { sin(x)/x, x not = 0
          { 1,         x = 0

and THIS function is defined and continuous at x=0. In other words, f(x) has a removable singularity; g(x) has no singularity, because I have removed it!

So I think that f(x)= (x^2 - 1) / x - x + 1/x is defined at x = 0.
As a matter of fact simplifying it yields zero. Since it approaches
zero from both sides when you take the limit, I think it is both
defined and a removable singularity and you can and should do the
simplification. Trouble is no one agrees with me. They say f(0) is
undefined, who is right?

The function as written is not defined for x=0. You cannot evaluate
it when x=0; it involves two divisions by zero, which are undefined.
However, if you simplify it as you say (assuming x is not equal to
0), you find that it is equal to the function

  g(x) = 0, x not equal to 0

This is NOT the same function as

  h(x) = 0

Why not? Because in order for two functions to be the same function,
they must have the same domain. These two functions, h(x) and g(x)
(or h(x) and f(x)), are indeed equal for all values of x in the
domains of both; but h(x) has zero in its domain while f and g do
not.

To answer your question then, "they" are right.

How was that for a mathematical spanking?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Even or Odd or Constant function?

That was what I was trying to say here:

Of course you can write ƒ(x) = 0. But you also have to specify that the domain is

and not

. Whoever said you can’t probably meant that you can’t just write

and leave it there, but also have to indicate that the function is not defined at 0.

But you completely ignored me.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Even or Odd or Constant function?

Hi Jane;

But you completely ignored me.

For the sin of ignoring you I formally apologize and promise not to do it again.

For the unpardonable sin of having made you cry, may a large camel spider alight on my face and eat my eyeballs.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.