**Contents:**

- What are Even and Odd Functions?
- Properties of Odd and Even Functions
- Test for Even and Odd Functions
- Integration of Even and Odd Functions

## What are Even and Odd Functions?

**Even and odd functions **have special symmetries about the origin or y-axis.

- A function is even if it is symmetric about the vertical y-axis; if this is the case, f(-x) = f(x) for every x in the domain.
- A function is odd if it is symmetric about the origin. In this case, f(-x) = -f(x) for every x in the domain.

## Examples of Even and Odd Functions

The parabola f(x) = x^{2} is an **example of an even function**. As you can see in the image below, it is symmetric about the vertical y axis.

The function f(x) = x^{3}, on the other hand, is an **example of an odd function**. It is symmetric about the origin.

Since the definition of even and odd are not mutually exclusive, it is possible to have a function that is both even and odd. For such a function, both f(-x) = f(x) and f(-x)= -f(x) are true. Substituting those equations into each other, we find that, for these functions, f(x) = -f(x). There is only one way that can be true: if f(x) = 0. So the function** f(x) = 0 **is the one and only function that is both even and odd.

## Properties of Odd and Even Functions

Here are some basic properties of odd and even functions that are worth remembering:

- The sum of two odd functions is odd, and the sum of two even functions is even (note that this is
**not**the same as the rule for even and odd integers) - The sum of an even and an odd function is neither even nor odd unless one or both of them is actually equal to zero.
- The difference of two even functions is even; likewise, the difference of two odd functions is odd.
- The product of two even functions is another even function, and the product of two odd functions is an even function also.
- The product of an even function and an odd function is an odd function.

## Test for Even and Odd Functions

This simple test states:

A function y – f(x) is

evenif f(-x) = f(x)

A function y – f(x) isoddif f(-x) = f(x)

**Example question #1:** Is the function f(x) = x^{3} – x odd or even?

**Solution:**

Step 1: **Replace any “x” in the function with “-x”:**

f(-x) = (-x)^{3} – (-x)

Step 2: **Solve with algebra.**The goal is to get the function in a form that is either the original function, x^{3} – x, or its negative -(x^{3} – x),

- f(-x) = (-x)
^{3}– (-x) = - = -x
^{3}+ x

The result, -x^{3} + x, is the negative of the function so this function is odd.

**Example question #2:** Is the function f(x) = 2 + cos(x) odd or even?

**Solution:**

Step 1: **Replace any “x” in the function with “-x”:**

f(-x) = 2 + cos(-x)

Step 2: **Solve with algebra.**The goal is to get the function in a form that is either the original function, x^{3} – x, or its negative -(x^{3} – x),

- f(-x) = 2 + cos(x)*

The result, 2 + cos(x) is the original function, so this is an even function.

One of the Trigonometric identities for negatives states that -cos(x) = cos(x).

## Integration of Even and Odd Functions

The theorem of “Integration of Even and Odd Functions” is a way to find integrals for odd and even functions. It’s **a method that makes some challenging definite integrals easier to find. **In order to use it, you have to be evaluating a function over an interval that’s either symmetric about the origin or symmetric about the y-axis [1].

In order to use this theorem to simplify finding a definite integral:

**Your closed interval must be in the form [-a, a].**In other words, your interval must have the same number “a” like [-3, 3] or [-π, π].**You must have an even or odd function.**If you aren’t sure use the Test for Even and Odd Functions to check before continuing.

## Integration of Even and Odd Functions: Even Example

**1. Even functions Rule: **

**Example question:** Find the follow integral:

Step 1:

**Rewrite the integral to cover the positive half of the region**(shown in the above image). For this example, instead of integrating from -1 to 1, we’re integrating from 0 to 1:

Step 2: **Solve the integral** (I used Symbolab’s calculator):

Solving the integral, we get 8/7 ≈ 1.14285.

Step 3: **Multiply the solution from Step 2 by 2:**

8/7 * 2 = 16/7 ≈ 2.28571.

*That’s it!*

## 2. Odd functions Definite Integral Example

This is the *easiest definite integral you’ll ever calculate* (actually, you don’t even have to do any calculations…read on!).

The formula is:

This part of the theorem is easier than for even functions. It simply states that **if your function is symmetric about the origin for interval [-a, a], the definite integral is going to be zero.**

## References

Newcastle University Academic Skills Kit. Odd and Even Functions. Retrieved from https://internal.ncl.ac.uk/ask/numeracy-maths-statistics/core-mathematics/pure-maths/functions/odd-and-even-functions.html on August 10, 2019

Smith, Ken W. Elementary Functions Part 1: Lecture 1.4a, Symmetries of Functions. Retrieved from https://www.shsu.edu/~kws006/Precalculus/1.4_Function_Symmetries_files/1.4%20FunctionSymmetries%20slides%204to1.pdf on August 10, 2019.

Siktar, Joshua. Algebraic Analysis of Functions: Even and Odd Functions. Retrieved from https://opencurriculum.org/5507/even-and-odd-functions/ on August 10, 2019.

Maplesoft. Basic Functions and Relations: Math Apps. Even and Odd Functions. Maplesoft Online Help. Retrieved from https://www.maplesoft.com/support/help/Maple/view.aspx?path=MathApps%2FEvenAndOddFunctions on August 10, 2019

Hi. Your definitions of odd and even functions on this page: https://www.statisticshowto.com/even-and-odd-functions/ are incorrect.

That page says for even functions :

A function is even if it is symmetric about the vertical y-axis; if this is the case, f(-x) = x for every x in the domain.

It should be f(-x) = f(x)

For odd functions, the page says:

A function is odd if it is symmetric about the origin. In this case, f(-x) = x for every x in the domain.

It should be f(-x) = -f(x)

Thank you for the correction, Grant.