Even and Odd Functions?

Question

Must the product of two even functions always be even? examples?

Can anything be said about the product of two odd functions? reasons?

thanks for any help...i tend to rely on both description and example so if you can answer please provide both! THANKS!!!

Answer 1

There is an easy proof for the first statement. Use the definition of even function and the definition for the product of functions. Also the product of two odd functions is even, also an easy proof. So to look for examples, you just need to know what some even and odd functions are.

Answer 2

Although most of the above answers are good, I am not sure if they actually answer the question dealing with products of even and odd function. So, I'll give it a whirl.

The best definition of an even function (at least in my opinion) is that it can be represented as a series (even if it is infinite) if terms containing x to even exponents. Thus, it takes on the form:

f(x)=a(x^0)+b(x^2)+c(x^4) ... or a+b(x^2)+c(x^4) ... for some constants a,b,c.

The similar case can be made for odd functions. Now examining the product of two even functions:

f(x)*g(x)= [a+b(x^2)+c(x^4) ... ]*[d+e(x^2)+f(x^4) ... ]
=a*d+a*e(x^2)+a*f(x^4) ... +b(x^2)*d+b(x^2)*e(x^4)....
= ad+ae(x^2)+af(x^4) ... + bd(x^2) +be(x^6) ...

and well, it can get ugly, but you'll notice the exponents of x always stay even, the product of two even functions are even. As for odd functions...

h(x)*j(x)= [a(x^1)+b(x^3)+c(x^5) ... ]*[d(x^1)+e(x^3)+f(x^5) ... ]
= a(x)*d(x)+a(x)*e(x^3) ... +b(x^3)*d(x^1) +b(x^3)*e(x^3) ...
= ad(x^2)+ae(x^4) ... +bd(x^4) +be(x^6) ...

These exponents reduce to even exponents, so again, the product is always even.

Answer 3

Here's all I know about even and odd functions.

If it is an odd function, all exponents in the function MUST be odd. Example:
f(x) = x^5 + 2x^3

Also, if it is odd, it CANNOT have a constant.
f(x) = x^5 + 2x^3 + 3 IS NOT odd because the constant of +3
f(x) = x^5 + 2x^3 IS odd because all exponents are odd and there is NO constant.

I believe if it is even, then all exponents MUST be even, and there CAN be a constant.
f(x) = 2x^4 + 2x^2 + 3 or
f(x) = 2x^4 + 2x^2

If the function is neither, then it is anything that does not fit the above description.
f(x) = 2x^2 + x - 3
This is neither because x is the same as x^1 and 1 is odd. The exponent of 2 is even, therefore it can be neither even or odd.

Hope that helped... at least a little bit. :-)

Answer 4

The product of two even functions will always be even. x^2 and x^4 or any variation on that.

You can't really say anything about the product of two odd functions, because you don't know how they are odd. Making it simple and talking about two functions that pass through the y-axis at 0, one could run in the first quadrant and the third quadrant, and the other in the second quadrant and the fourth quadrant. (x)^3 and (-x)^3, for example.

Answer 5

This proof is done by a process called mathematical induction. However, an easier way to do this is to say that Let the first number be 2n. This will always be an even number because any number multiplied by 2 will be even. Then let the second number be 2k, with n and k being 2 different numbers. So when you multiply these numbers together you get (2n)(2k) = 2 * 2 * n * k. No matter what n * k is, since it will be multiplied by 2, it will be even as mentioned above. Hence the product of two even numbers is an even number.

Do the same process with odd numbers: Let 1st # = 2k + 1, and let the second number be 2n + 1. With n and k being different. (2k + 1)(2n + 1) = 4nk + 2k + 2n + 1. 4nk, 2k, 2n will all be even. And even numbers added together will always be even. But the +1 at the end makes it an odd number.

Answer 6

The definition of an even function f is that for all x, f(x) = f(-x). The definition of fg, the multiplicative product of two functions, is fg(x) = f(x)g(x). If f and g are even, then this is the same as f(-x)g(-x). What is this fg of, and what does that say about the function fg?

Do something similar if both f and g are odd; i.e., f(x) = -f(-x) and g(x) = -g(-x) for all x. Here you multiply two minuses together to get a plus, so this says something about fg in this case.

Answer 7

a function f(x) is said to be even if f(x)=f(-x)
let us take a simple case of f(x)=x^2
f(-x) will also be x^2 and so f(x)=f(-x) and
so the function(x)=x^2 is even
if f(-x)=-f(x) then the function is odd
again taking simple example of f(x)=x
f(-x)=-x=-f(x) and so f(x)=x is an odd function
if f(-x) is neither f(x) nor -f(x) then the function is
neither odd nor even
Here is a list of common even and odd functions:
even functions x^n with n even,polynomials with only even degree terms,cosx,secx
odd functions x^n with n odd,polynomials with only 0dd degree terms,sinx,cosecx
neither odd nor even are general polynomials,shifted trigonometric functions such as sin(3+x)
as far as the products are concerned both the products will be
even even into even or odd into odd

Answer 8

like firat_c siad, both products are even.

even means: f(-x) = f(x)
odd means: f(-x) = -f(x)

so let f & g be even functions:

f(-x)g(-x) = f(x)g(x), this is even

let f & g be odd functions:

f(-x)g(-x) = (-f(x))(-g(x)) = f(x)g(x), this is even

some examples of each have been given already

Answer 9

Both products yield even functions:

1) If both f and g are even then f(x)=f(-x) and g(x)=g(-x).
then
f(-x)g(-x)=f(x)g(x)
and hence fg is even.

2) If both f and g are odd, then
f(x)=-f(x) and g(-x)=-g(x),
therefore
f(-x)g(-x)=(-f(x))(-g(x))
=f(x)g(x).