# Even and Odd Functions

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https://algebrica.org/even-and-odd-functions/
Fetched from algebrica.org post 15387; source modified 2025-12-06T17:55:38.

## Behavior of a function

When analyzing the behavior of a [function](../functions.md), it is useful to investigate whether the function exhibits symmetry with respect to the coordinate axes. In this context, functions can be classified as **even**, showing symmetry with respect to the \\(y\\)-axis, or **odd**, exhibiting symmetry with respect to the origin. In general, a function can be:

-   even
-   odd
-   neither even nor odd

## Even function

More specifically, suppose we have a function \\( f(x): \\mathbb{R} \\rightarrow \\mathbb{R} \\), and let \\( D \\subseteq \\mathbb{R} \\) be its [domain](../determining-the-domain-of-a-function.md). The function \\( f \\) is said to be **even** if the following condition holds:

\\[
f(x) = f(-x) \\quad \\text{for all } x \\in D
\\]

![](https://algebrica.org/wp-content/uploads/resources/images/even-odd-functions-1.png)

As shown in the figure, the function \\( f(x) = x^2 \\) is a [parabola](../parabola.md) symmetric with respect to the \\(y\\)-axis. In general, functions of the form \\( f(x) = x^4 \\), \\( x^6 \\), or more generally \\( x^{2n} \\), where the exponent is even, are examples of even functions.

![](https://algebrica.org/wp-content/uploads/resources/images/even-odd-functions-2-1.png)

Another example of an even function is the [cosine function](../cosine-function.md). It is a periodic function with period \\( 2\\pi \\), and its graph is symmetric with respect to the \\(y\\)-axis. In fact, it is easy to verify that:

\\[
\\cos(\\pi) = \\cos(-\\pi) = -1
\\]

Another even function is the [absolute value function](../absolute-value-function.md).

* * *

More generally, when considering the family of functions of the form \\(f(x) = x^{n}\\) with \\(n \\in \\mathbb{N},\\) the parity of the function is entirely determined by the exponent: the function behaves as an even function whenever \\(n\\) is an even [integer](../integers.md), whereas it behaves as an odd function whenever \\(n\\) is odd.

## Definite integral of even function

One of the useful consequences of a function being even is the simplification it allows in [definite integrals](../definite-integrals.md) over symmetric intervals. If \\( f(x) \\) is a [continuous](../continuous-functions.md) and even function, then its graph is symmetric with respect to the \\(y\\)-axis.

This symmetry directly influences how we evaluate definite integrals over intervals of the form \\(\[ -a, a \]\\). Specifically, the following identity holds:

\\[
\\int\_{-a}^{a} f(x),dx = 2\\int\_0^a f(x),dx
\\]

![](https://algebrica.org/wp-content/uploads/resources/images/definite-integrals-5.png)

That is, the total area under the curve from \\(-a\\) to \\(a\\) is simply twice the area from 0 to (a). This works because the portion of the graph on the negative side of the \\(x\\)-axis is a mirror image of the positive side, and contributes the same value to the integral.

## Odd function

Suppose we have a function \\( f(x): \\mathbb{R} \\rightarrow \\mathbb{R} \\), and let \\( D \\subseteq \\mathbb{R} \\) be its domain. The function \\( f \\) is said to be **odd** if the following condition holds:

\\[
f(-x) = -f(x) \\quad \\text{for all } x \\in D
\\]

![](https://algebrica.org/wp-content/uploads/resources/images/even-odd-functions-3.png)

As shown in the figure, the function \\( f(x) = x^3 \\) is symmetric with respect to the origin. Functions of the form \\( f(x) = x^3 \\), \\( x^5 \\), or more generally \\( x^{2n+1} \\), where the exponent is odd, are examples of odd functions.

![](https://algebrica.org/wp-content/uploads/resources/images/even-odd-functions-4-1.png)

Another example of an odd function is the [sine function](../sine-function.md). It is a periodic function with period \\( 2\\pi \\), and its graph is symmetric with respect to the origin. In fact, it is easy to verify that:

\\[
\\sin(-\\pi) = -\\sin(\\pi) = 0
\\]

## Definite integral of odd function

In the case of an odd function, the area between \\( \[-a, 0\] \\) is equal in magnitude but opposite in sign to the area between \\( \[0, a\] \\). Therefore, the definite integral is equal to:

\\[
\\int\_{-a}^{a} f(x) \\, dx = 0
\\]

![](https://algebrica.org/wp-content/uploads/resources/images/definite-integrals-6.png)

In both situations, the area enclosed between the graph of \\( f(x) \\) and the \\( x \\)-axis over the interval \\( \[-a, a\] \\) is given by:

\\[
S = \\int\_{0}^{a} |f(x)| \\, dx
\\]

## The only function that is both even and odd

The function \\( f(x) = 0 \\) is the only function that is both even and odd, because it satisfies both \\( f(-x) = f(x) \\) and \\( f(-x) = -f(x) \\) for all \\( x \\in \\mathbb{R} \\). In fact, if a function were to be both even and odd, we would have:

-   \\(f(-x) = f(x)\\) when the function is even.
-   \\(f(-x) = -f(x)\\) when the function is odd.

Therefore, the zero function is the unique case that satisfies both properties.

## Properties

-   The sum of two even functions is even.
-   The product of an even function by a constant is even.
-   The product of two even functions is an even function.
-   The [derivative](../derivatives.md) of an even function is an odd function.
-   The sum of two odd functions is odd.
-   The product of an odd function by a constant is odd.
-   The product of two odd functions is an even function.
-   The derivative of an odd function is an even function.
-   The product of an even function and an odd function is an odd function.