# Proving even and odd functions

## Main Question or Discussion Point

Can someone prove even and odd fucntions for me not through examples but by actually proving them?

Thanks

Well how do you define even and how do you define odd functions?

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.

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?

Is that a proof though?

Yes... you are taking a definition and using it. What's a proof in your opinion?

You are defining even and odd functions to have those properties. There is no need for a proof.

IS there a way to prove an even and odd function?

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.

symbolipoint
Homework Helper
Gold Member
Can someone prove even and odd fucntions 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.

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.

so could an axiom be f(x)=|x|?

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

symbolipoint
Homework Helper
Gold Member
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?

Mentallic
Homework Helper
What about proving that if an even function by definition being $$f(x)=f(-x)$$ is a reflection in the y-axis and an odd function defined as $$f(-x)=-f(x)$$ has symmetry about the origin?

HallsofIvy
Homework Helper
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?