why an odd function always start with zero?

 

Asad1,

why an odd function always start with zero?

What do you mean? Be definition, the function starts in the negative region of the x-axis for both odd and even functions. Perhaps an example is in order to understand what you are asking.

Ratch

 

Odd function always = 0 at zero: f(0) = 0

 

Alexei Smirnov,

Odd function always = 0 at zero: f(0) = 0

f(x) = 1/x is a odd function which is not defined at 0.

Ratch

 

Alexei Smirnov,



f(x) = 1/x is a odd function which is not defined at 0.

Ratch

Ok, ok
f(-x)=-f(x)...
only continuous function=0...

 

Actually i need a mathematical and analytical prove that an odd signal have zero at the origin. Like sine is an odd function and it starts with Zero, The question is that why an odd function always get start from 0.

 

Actually i need a mathematical and analytical prove that that an odd signal have zero at the origin. Like sine is an odd function and it starts with Zero, The question is that why an odd function always get start from 0.

f(-x)=-f(x); (definition of odd function)
f(-x)+f(x)=0;
if x=0 ->
f(-0)+f(0)=0;
2*f(0)=0;
f(0)=0;
But, the function should be defined(?) at 0. Don't know what to do with 1/x, log(x), ...

 

Asad1,

Actually i need a mathematical and analytical prove that that an odd signal have zero at the origin. Like sine is an odd function and it starts with Zero, The question is that why an odd function always get start from 0.

Even functions are mirror images with respect to only the y-axis. Odd junctions are upside mirror images of with respect to both the x and y axis. Therefore odd functions reference both the x and y axis, which intersect at the origin. It is wrong for you say "start" at zero when you mean referenced at the origin. Functions start from the minus x-axis and end at plus x-axis. Furthermore, you are asking to prove a definition, Definitions cannot be proven.

Ratch

 

Thank you soooooooooooooo much

 

Furthermore, you are asking to prove a definition, Definitions cannot be proven.
Ratch

Wait a second, being pedantic...definition of odd function is
f(-x) = -f(x).
This does not necessarily mean f(0) = 0, so it can (must) be proven...

 

Alexei Smirnov,

Wait a second, being pedantic...definition of odd function is
f(-x) = -f(x).
This does not necessarily mean f(0) = 0, so it can (must) be proven...

The definition of a odd function does not have to be proven. There are odd functions such as the one I submitted earlier f(x)=1/x which do not pass through the origin. The plot follows its reflection with both the y and the x axis.

Ratch

 

In general Ratch has it correct.

There is no requirement for f(0) = 0.

You simply have to lift any even or odd function by adding a constant eg f(x) =sin(x) +1.

There are other pitfalls though so I have appended some sketches based on straight lines.

Many textbooks state that you can always write any function as the sum of an even plus an odd function and great use is made of this in electrical engineering. However this statement is only strictly true of certain classes of function, in particular polynomials and trigonometric functions.

I have also displayed one very important function which is both odd and even.

 

Many textbooks state that you can always write any function as the sum of an even plus an odd function and great use is made of this in electrical engineering. However this statement is only strictly true of certain classes of function, in particular polynomials and trigonometric functions.

Do you mean that not any function can be represented as a sum of even and odd functions? I think that's not true, and can easily be proven.

 

Consider

f(x) =
\(\frac{1}{2}\)[f(x) + f(-x)] + \(\frac{1}{2}\)[f(x) - f(-x)]

an even function plus an odd function.

 

Exactly.

Did I misunderstand your statement:

However this statement is only strictly true of certain classes of function, in particular polynomials and trigonometric functions.

, in other words "not ALL functions can be represented by sum of odd and even?

 

No you did not misunderstand.

I gave you a way of deconstructing f(x) into odd and even components, so long as f(-x) is defined.
Of course f(-x) is not defined for all functions, eg square root.