Even and Odd Functions

The sine function and all of its Taylor polynomials are odd functions. This image shows $\sin(x)$ and its Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13.

The cosine function and all of its Taylor polynomials are even functions. This image shows $\cos(x)$ and its Taylor approximation of degree 4.

Evenness and oddness are generally considered for real function]]s, that is real-valued functions of a real variable. However, the concepts may be more generally defined for functions whose domain and codomain both have a notion of additive inverse. This includes abelian groups, all rings, all fields, and all vector spaces. Thus, for example, a real function could be odd or even (or neither), as could a complex-valued function of a vector variable, and so on.

The given examples are real functions, to illustrate the symmetry of their graphs.

Even functions

$f(x)=x^{2}$ is an example of an even function.

Let f be a real-valued function of a real variable. Then f is even if the following equation holds for all x such that x and −x in the domain of f:

$f(x)=f(-x)$

or equivalently if the following equation holds for all such x:

$f(x)-f(-x)=0.$

Geometrically, the graph of an even function is symmetric with respect to the y-axis, meaning that its graph remains unchanged after reflection about the y-axis.

Examples of even functions are:

• The absolute value $y=|x|$
• $y=x^{2}$
• $y=x^{4}$
• The cosine function $y=\cos {x}$
• The hyperbolic cosine function $y=\cosh {x}$

Odd functions

$f(x)=x^{3}$ is an example of an odd function.

Again, let f be a real-valued function of a real variable. Then f is odd if the following equation holds for all x such that x and −x are in the domain of f:

$-f(x)=f(-x)$

or equivalently if the following equation holds for all such x:

$f(x)+f(-x)=0.$

Geometrically, the graph of an odd function has rotational symmetry with respect to the origin, meaning that its graph remains unchanged after rotation of 180 degrees about the origin.

Examples of odd functions are:

• The identity function $y=x,$
• $y=x^{3},$
• The sine function $y=\sin {x},$
• The hyperbolic sine function $y=\sinh {x},$

$f(x)=x^{3}+1$ is neither even nor odd.

Basic properties

Uniqueness

• If a function is both even and odd, it is equal to 0 everywhere it is defined.
• If a function is odd, the absolute value of that function is an even function.

Addition and subtraction

• The sum of two even functions is even.
• The sum of two odd functions is odd.
• The difference between two odd functions is odd.
• The difference between two even functions is even.
• The sum of an even and odd function is not even or odd, unless one of the functions is equal to zero over the given domain.

Multiplication and division

• The product of two even functions is an even function.
  • That implies that product of any number of even functions is an even function as well.
• 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.

Composition

• The composition of two even functions is even.
• The composition of two odd functions is odd.
• The composition of an even function and an odd function is even.
• The composition of any function with an even function is even (but not vice versa).

Even–odd decomposition

Every function may be uniquely decomposed as the sum of an even and an odd function, which are called respectively the even part and the odd part of the function; if one defines

$f_{\text{e}}(x)={\frac {f(x)+f(-x)}{2}}$

and

$f_{\text{o}}(x)={\frac {f(x)-f(-x)}{2}}$

then $f_{\text{e}}$ is even, $f_{\text{o}}$ is odd, and

$f(x)=f_{\text{e}}(x)+f_{\text{o}}(x).$

Conversely, if

$f(x)=g(x)+h(x),$

where g is even and h is odd, then $g=f_{\text{e}}$ and $h=f_{\text{o}},$ since

${\begin{aligned}2f_{\text{e}}(x)&=f(x)+f(-x)=g(x)+g(-x)+h(x)+h(-x)=2g(x),\\2f_{\text{o}}(x)&=f(x)-f(-x)=g(x)-g(-x)+h(x)-h(-x)=2h(x).\end{aligned}}$

For example, the hyperbolic cosine and the hyperbolic sine may be regarded as the even and odd parts of the exponential function, as the first one is an even function, the second one is odd, and

$e^{x}=\underbrace {\cosh(x)} _{f_{\text{e}}(x)}+\underbrace {\sinh(x)} _{f_{\text{o}}(x)}$.

Further algebraic properties

• Any linear combination of even functions is even, and the even functions form a vector space over the reals. Similarly, any linear combination of odd functions is odd, and the odd functions also form a vector space over the reals. In fact, the vector space of all real functions is the direct sum of the subspaces of even and odd functions. This is a more abstract way of expressing the property in the preceding section.
  • The space of functions can be considered a graded algebra over the real numbers by this property, as well as some of those above.

• The even functions form a commutative algebra over the reals. However, the odd functions do not form an algebra over the reals, as they are not closed under multiplication.

Analytic properties

A function's being odd or even does not imply differentiability, or even continuity. For example, the Dirichlet function is even, but is nowhere continuous.

In the following, properties involving derivatives, Fourier series, Taylor series, and so on suppose that these concepts are defined of the functions that are considered.

Basic analytic properties

• The derivative of an even function is odd.
• The derivative of an odd function is even.
• The integral of an odd function from −A to +A is zero (where A is finite, and the function has no vertical asymptotes between −A and A). For an odd function that is integrable over a symmetric interval, e.g. $[-A,A]$, the result of the integral over that interval is zero; that is

• $\int _{-A}^{A}f(x)\,dx=0$.

• The integral of an even function from −A to +A is twice the integral from 0 to +A (where A is finite, and the function has no vertical asymptotes between −A and A. This also holds true when A is infinite, but only if the integral converges); that is

• $\int _{-A}^{A}f(x)\,dx=2\int _{0}^{A}f(x)\,dx$.

Resources

• Single Transformations of Functions (MAT1073)

Licensing

Content obtained and/or adapted from:

• Even and odd functions, Wikipedia under a CC BY-SA license