What Are Even and Odd Functions?

[eax]

Is this right?

An even function has this property
f(x)=f(-x)
and an odd function has this property
-f(x) = f(-x)

 

[hypermorphism]

Yes. Note that many functions are neither even nor odd.

 

[eax]

Yes. Note that many functions are neither even nor odd.

Thanks! I just had a test, and one question said to give an example of an odd function and another to prove a function is odd. I looked at the "prove" question and guessed correctly :).

 

[Robokapp]

Thanks! I just had a test, and one question said to give an example of an odd function and another to prove a function is odd. I looked at the "prove" question and guessed correctly :).

symetry can be of 4 types:

on the x-axis (like y^2-x^2=0 or the equation of an elipse etc)
on the y-axis (like y=x^2 or a function with even degree - hence it's an "even" function)
on the origin (meaning that if point (1,5) and (2,10) belong to it so must points (-1,-5) and (-2,-10)...in other words it is copied inversed in the negative direction)
on the y=x axis (like any function and it's inverse or like y=x+1 and y=x-1 for example)

i think the "odd" function is that symetric on orrigin i don't remember.

there are functions that are not symetric to anything. example: e^(-x)Sin(x^3). it would be just a function oscilating back and forth across the x-axis and ending up in a horisontal asymptote at y=0.

 

[fourier jr]

odd functions have rotational symmetry; even functions have reflectional(?) symmetry

 

[HallsofIvy]

The reason for the names is that every polynomial in x, having only even powers of x, is an even function, every polynomial in x, having only odd powers of x, is an odd function. Most polynomials, have both even and odd powers are neither.

OF course, "even" and "odd" applies to other functions as well: sin(x) is an odd function and cos(x) is an even function.

Given any function, f(x), we can define the "even" and "odd" parts of f by
$$f_{odd}(x)= \frac{f(x)- f(-x)}{2}$$
$$f_{even}(x)= \frac{f(x)+ f(-x)}{2}$$

If f(x)= e^x, which is itself neither even nor odd, we get
f_odd(x)= sinh(x) and f_even= cosh(x).