Even and Odd Functions

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Source: algebrica.org - CC BY-NC 4.0 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, 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. The function $f$ is said to be even if the following condition holds:

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

As shown in the figure, the function $f(x) = x^2$ is a parabola 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.

Another example of an even function is the cosine function. 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.

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, 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 over symmetric intervals. If $f(x)$ is a continuous 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

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

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.

Another example of an odd function is the sine function. 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

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 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.