Symmetry functions and integration

Question

I have this:

Case 1)

If f is a pair function $f(-x)=f(x)$ then $\int_{-a}^a f(x)dx=2\int_0^af(x)dx$

Case 2)

If $f$ is a inpair function $f(-x)=-f(x)$ then $\int_{-a}^a f(x)dx=0$

I understand the reasons for the case 1 to be double area and for case 2 to be zero, but, I'll be grateful if some one can tell me a little more of this symmetry aspect

How can I realize when and where this symmetry exist in a function Particularly will be helpful if someone explain how to interpret $f(-x)=-f(x)$ and $f(-x)=f(x)$ I'd to understand better what do they imply

Answer 1

How can I realize when and where this symmetry exist in a function?

You can recognize symmetric functions by knowing basic examples and understanding how these behave under common combinations.

1. The most basic examples of even functions $f(x)=f(-x)$ are the monomials with even exponent. For instance: $1=x^0, x^2, x^4$ and so on. The function $f(x)=x^2$ is even since $$f(-x)=(-x)^2=(-1)^2x^2=x^2=f(x).$$ Examples of odd functions $f(x)=-f(-x)$ are given by the monomials with odd exponent: $x,x^3,x^5,\cdots$. Two other examples basic examples of functions with symmetry are sine and cosine. By the Taylor expansion $$\sin(x) = \sum_{k=0}^{\infty} \frac{(-1)^n}{(2n+1)!}x^{2n+1} = x-\frac{x^3}{3!}+\frac{x^5}{5!}+\cdots$$ we suspect that sine is odd since it consists of only odd powers of $x$. You should verify this. Alternatively, the Taylor expansion for cosine indicates that it is an even function.

2. Now that you have a library of functions exhibiting symmetry, you can ask:

   • What happens if I add two such functions?
   • What if I multiply a symmetric function by a constant?
   • What if I multiply two such functions?
   • What happens if I divide two such functions?
   • What if I compose two such functions?

I won't answer all of these questions, but here's an example. Let's prove that the product of an even function with an odd function is odd. Let $f$ be an even function, that is, $f(-x)=f(x)$, and $g$ satisfy $g(-x)=-g(x)$. Then we want to show that $x \mapsto f(x)g(x)$ is an odd function. Simply compute: $$f(-x)g(-x) = f(x)g(-x) = f(x)[-g(x)] = -f(x)g(x).$$ So the function $fg$ is odd!

As a concrete example, is the function $x^2\sin(x)$ even or odd? What about $\tan(x)$? What about $\cos(\tan(x))$ (for $x$ in the domain of the function)? Look at the graph, then try to prove it algebraically.

Answer 2

Algebraically,

A function is odd if $f(-x)=-f(x)$ and a function is even when $f(-x)=f(x)$.

For example, if $f(x)=x^3$, then $f(-x)=(-x)^3=-x^3$, which is $-f(x)$. This is our original function multiplied by $-1$. That means we can say this is an odd function.

Example for an even function could be $f(x)=3x^2$, because $f(-x)=3(-x)^2=3x^2$, which is our original function, $f(x).$ This means we can say this is an even function.

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Geometrically,

A function $f$ is even if the graph of $f$ is symmetric with respect to the $y$-axis.

A function $f$ is odd if the graph of $f$ is symmetric with respect to the origin.

Answer 3

There are many reasons why we call those even and odd functions. One is that the taylor series of an even function only includes even powers whereas the Taylor series of an odd function only includes odd power.

But to recognize when and what type of symmetry there is, the most useful property is that they are called even/odd because they often act similar to even/odd numbers:

• The sum of two even functions is even, and any constant multiple of an even function is even.
• The sum of two odd functions is odd, and any constant multiple of an odd function is odd.
• The difference between two odd functions is odd.
• The difference between two even functions is even.
• The product of two even functions is an even function.
• The product of two odd functions is an even function.
• The product of an even function and an odd function is an odd function.
• The quotient of two even functions is an even function.
• The quotient of two odd functions is an even function.
• The quotient of an even function and an odd function is an odd function.

With that in mind, you need to have a couple basic even/odd functions in your sleeve. $x^n$ is even for $n$ even and odd for $n$ odd (even for negative integers). the function $\cos$ is even while $\sin$ is odd. What would $\tan$ be? The absolute value $|x|$ is even.

When looking at some functions $f(x)=g(x)/h(x)$, use the above properties to determine the "parity" of $g(x)$ and $h(x)$ and then check their quotient.

Note however that some functions are neither even nor odd, for example $x^3-x+1$.