Proving even and odd functions

In summary, an even function is defined as a function whose graph is symmetric with respect to the y-axis, or algebraically, when f(-x) = f(x) for all x in the domain of f. An odd function is defined as a function whose graph is symmetric with respect to the origin, or algebraically, when f(-x) = -f(x) for all x in the domain of f. These definitions can be used to prove the identity of even and odd functions by plugging in specific values and observing the result. In order to prove the symmetry of an even function with respect to the y-axis and an odd function with respect to the origin, one must first define the terms "reflection in the y-axis"
  • #1
darthxepher
56
0
Can someone prove even and odd functions for me not through examples but by actually proving them?

Thanks
 
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  • #2
Well how do you define even and how do you define odd functions?
 
  • #3
DEFINITION. A function f is even if the graph of f is symmetric with respect to the y-axis. Algebraically, f is even if and only if f(-x) = f(x) for all x in the domain of f.

A function f is odd if the graph of f is symmetric with respect to the origin. Algebraically, f is odd if and only if f(-x) = -f(x) for all x in the domain of f

I need an algebraic proof using angles and algebra... its for my trig class.
 
  • #4
Angles and algebra? Why don't you do it by plugging -x in? Like you said if f(-x) = f(x) then f is even if f(-x) = -f(x) then f is odd. Where are you getting lost?
 
  • #5
Is that a proof though?
 
  • #6
Yes... you are taking a definition and using it. What's a proof in your opinion?
 
  • #7
You are defining even and odd functions to have those properties. There is no need for a proof.
 
  • #8
IS there a way to prove an even and odd function?
 
  • #9
How can you prove in general what an even or odd function is? First you have to define what you mean by that, but once you do that, there is no need for a proof.
 
  • #10
darthxepher said:
Can someone prove even and odd functions for me not through examples but by actually proving them?

Thanks

Start with a specific function and test it for identity of evenness and oddness according to the definition for even and odd functions.
 
  • #11
I'm not sure you understand what a proof is. You have a bunch of axioms, and you use those to arrive from part A to part B. Your axiom is what an even and what an odd function are. Using them, you show that some functions are even, some are odd, some are neither.
 
  • #12
so could an axiom be f(x)=|x|?
 
  • #13
What axiom would that be? If you are trying to DEFINE an absolute value function then you can say, f(x) = |x| means f(x) = x for x >= 0 and -x for x <= 0
 
  • #14
darthxepher said:
so could an axiom be f(x)=|x|?

USE the Definitions of EVEN functions and ODD functions. Does one statement or the other become an identity?

Check:
|-x|=|x|

Check:
|-x|=-|x|

It one of those or both of those or neither of those true? What is the meaning?
 
  • #15
What about proving that if an even function by definition being [tex]f(x)=f(-x)[/tex] is a reflection in the y-axis and an odd function defined as [tex]f(-x)=-f(x)[/tex] has symmetry about the origin?
 
  • #16
Then you have to start with definitions of "reflection in the y-axis" and "symmetry about the origin"! It should be obvious that you can't prove anything about "X" if you don't know what "X" means and, further, math definitions are "working definitions"- you use the precise words of the definitions themselves in working with the concepts.

The reflection of (x,y) in the y-axis is the point (-x, y). If f is an even function and y= f(x), what are (x, y) and (-x, y) in terms of the graph of f? Are they both on the graph?

The point "symmetric" to (x, y) in the origin is (-x, -y). If f is an odd function and y= f(x), what are (x, y) and (-x, -y) in terms of the graph of f? Are they both on the graph?
 

1. What is an even function?

An even function is a type of mathematical function where the output remains the same when the input is replaced with its negative value. In other words, f(x) = f(-x). Even functions are symmetric with respect to the y-axis, meaning they have a line of symmetry at y = 0.

2. What is an odd function?

An odd function is a type of mathematical function where the output becomes the negative value when the input is replaced with its negative value. In other words, f(x) = -f(-x). Odd functions are symmetric with respect to the origin, meaning they have a point of symmetry at the origin (0,0).

3. How can I prove if a function is even or odd?

To prove if a function is even, we substitute -x for x in the function and simplify. If the resulting function is the same as the original function, then it is even. To prove if a function is odd, we substitute -x for x in the function and simplify. If the resulting function is the negative of the original function, then it is odd.

4. Why is it important to know if a function is even or odd?

Knowing if a function is even or odd can help us solve problems involving symmetry and simplifying expressions. It also allows us to use properties and theorems specific to even and odd functions, such as the fact that the product of two even functions is even, and the product of an even and an odd function is odd.

5. Can a function be both even and odd?

No, a function cannot be both even and odd. This is because even functions have a line of symmetry at y = 0, while odd functions have a point of symmetry at the origin (0,0). The only function that satisfies both of these conditions is the constant function f(x) = 0, which is neither even nor odd.

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