Question about composing functions

  • Thread starter CausativeAgent
  • Start date
  • Tags
    Functions
In summary: So we can conclude that F(G(X)) is even if F is even and G is odd. This is the proof we were looking for.In summary, when considering the function F(G(X)), if F(X) is even and G(X) is either even or odd, the resulting function will always be even. This is because the product of two even numbers or an odd number and an even number will always result in an even number. Additionally, by using the definitions of even and odd functions, we can prove that F(G(X)) is even without needing to know anything specific about F or G.
  • #1
CausativeAgent
18
0
I am trying to give an explination or proof as to why F(G(X)) is always an even function if F(X) is even and G(X) is either even or odd. I understand why in simple situations such as when the resulting function is (X^3)^2 or (X^2)^2. It's because the product of two even numbers OR an odd number and a even number are both even.

But what if the resulting function is something more complicated such as
(X^3+X^5)^2. At first I thought that the outside function determines the symmetry, but then I tried
(X^2+X^4)^3, where the outside function is odd, and realized that this function is also even. I am stumped on how to prove, or even put into words, the reason for this function behavior.
 
Mathematics news on Phys.org
  • #2
Use the definitions of even and odd functions. In this case, viewing details yields little insight.
In other words, simply find the sign of F(G(-x)) vs. F(G(x)) using only the defining behavior of even and odd functions.
 
  • #3
What are the properties of odd and even functions, in general?

Wow, super fast post by slider!
 
  • #4
think about what it means for a function to be even, by definition of even function.

A function f is even if f(x)=f(-x) for all x.

Consider f(g(-x)).
we have two cases, g is even or g is odd, we will look at each individually.

if g is even then f(g(-x)) = f(g(x)) so f(g(x)) is even by definition of even.

Consider the case where g is odd (i.e. g(-x) = -g(x)). If g is odd we know (f(g(-x)) = f(-g(x)). but we know f is even so f(-x)=f(x) FOR ANY X (including g(x)). so f(-g(x)) = f(g(x)). But this means f(g(-x)) = f(g(x)) thus f(g(x)) is even.
 
  • #5
Or you could just tell him the answer! This is probably a homework question (if not, it sure looks like one) but even if it's not, it's better to let someone think about the question and the definitions of the functions, than to simply tell him what to do!
 
  • #6
slider142 said:
Use the definitions of even and odd functions. In this case, viewing details yields little insight.
In other words, simply find the sign of F(G(-x)) vs. F(G(x)) using only the defining behavior of even and odd functions.


In an even function all points are reflected across the Y axis. In other words for every point (X,Y) there is a corresponding point (-X,Y). The value of Y does not change when X is replaced by -X, i.e. F(-x)= F(X)

An odd function has an opposite value for Y when X is replaced by -x, i.e.
G(-x) = -G(x).
--------------------------------------
So here are the most basic properties of even and odd functions,F is even and G is odd:

F(-X) = F(X)

G(-X)=-G(X)

F(G(X)) = X

F(G(-X)) = X

Is this what you mean?
 
  • #7
JonF said:
think about what it means for a function to be even, by definition of even function.

A function f is even if f(x)=f(-x) for all x.

Consider f(g(-x)).
we have two cases, g is even or g is odd, we will look at each individually.

if g is even then f(g(-x)) = f(g(x)) so f(g(x)) is even by definition of even.

Consider the case where g is odd (i.e. g(-x) = -g(x)). If g is odd we know (f(g(-x)) = f(-g(x)). but we know f is even so f(-x)=f(x) FOR ANY X (including g(x)). so f(-g(x)) = f(g(x)). But this means f(g(-x)) = f(g(x)) thus f(g(x)) is even.

Thanks Jon that makes sense. So the first step is to realize that an opposite value of X on the inside odd function will yield an opposite value for that whole function G(X), then mentally replace -G(X) with -X and remember that for an even function F(-X)=F(X). That is so cool.
 
  • #8
CausativeAgent said:
In an even function all points are reflected across the Y axis. In other words for every point (X,Y) there is a corresponding point (-X,Y). The value of Y does not change when X is replaced by -X, i.e. F(-x)= F(X)

An odd function has an opposite value for Y when X is replaced by -x, i.e.
G(-x) = -G(x).
--------------------------------------
So here are the most basic properties of even and odd functions,F is even and G is odd:

F(-X) = F(X)

G(-X)=-G(X)

Up to that part, it's good.

F(G(X)) = X

This is only true if F is the inverse function of G. We don't know anything specific about F(G(X)) except that F is even and G is odd. But using the definitions we know, we can simplify the expression F(G(-X)), as Jon has done.
 

1. What is composing functions?

Composing functions is the process of combining two or more functions to create a new function. This new function will perform the operations of the individual functions in a specific order.

2. Why is composing functions useful?

Composing functions allows us to break down complex problems into smaller, more manageable parts. It also allows us to reuse functions that we have already written, making our code more efficient and organized.

3. How do you compose functions?

To compose functions, you simply take the output of one function and use it as the input for another function. This can be done in many programming languages, such as JavaScript, using the function composition operator (e.g. f(g(x))).

4. Can any two functions be composed together?

In most cases, yes. However, there are certain cases where functions may not be compatible for composition, such as if they have conflicting input or output types. It is important to carefully consider the compatibility of functions before composing them.

5. Are there any limitations to composing functions?

One limitation of composing functions is that the order in which the functions are composed matters. This means that changing the order of composition can result in a different output. Additionally, composing too many functions can make the code more complex and difficult to debug.

Similar threads

  • General Math
Replies
28
Views
3K
  • General Math
Replies
9
Views
2K
Replies
4
Views
825
  • General Math
Replies
2
Views
686
Replies
17
Views
2K
Replies
6
Views
1K
Replies
4
Views
284
Replies
13
Views
903
  • General Math
Replies
12
Views
1K
Back
Top