Even and Odd Functions
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:
\[f ( x ) = f ( - x ) \text{for all} x \in D\]
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 $x^{2 n}$, 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:
\[\int_{- a}^{a} f ( x ) , d x = 2 \int_{0}^{a} f ( x ) , d x\]
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:
\[f ( - x ) = - f ( x ) \text{for all} x \in D\]
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 $x^{2 n + 1}$, 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:
\[\int_{- a}^{a} f ( x ) d x = 0\]
In both situations, the area enclosed between the graph of $f ( x )$ and the $x$-axis over the interval $[ - a , a ]$ is given by:
\[S = \int_{0}^{a} | f ( x ) | d x\]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.