Even and Odd Functions
An even function satisfies f(−x) = f(x) and has a graph symmetric about the y-axis, while an odd function satisfies f(−x) = −f(x) and has a graph symmetric about the origin. In brief: f(x) = x2 is even because (−3)2 = 32 = 9, and f(x) = x3 is odd because (−2)3 = −8 = −(23). Many functions are neither.
Symmetry saves work. If you know a function is even or odd, you only need to study it for x ≥ 0 and then reflect the result. The idea is also used in integration, in Fourier series and in physics. This guide explains how to test symmetry algebraically, how it looks on a graph and how even and odd functions combine.
What are even and odd functions?
A function f is even if f(−x) = f(x) for every x in its domain, and odd if f(−x) = −f(x) for every x in the domain. In both cases, the domain must be symmetric about 0: if x is in the domain, so must be −x. The function f(x) = √x has domain [0, ∞), which is not symmetric, so it is neither even nor odd.
The names come from power functions: x2, x4 and other even powers are even functions, while x, x3 and other odd powers are odd functions. The only function that is both even and odd is the zero function f(x) = 0.
How do you test a function algebraically?
- Check that the domain is symmetric about 0.
- Compute f(−x) by replacing every x with (−x).
- Simplify and compare with f(x) and with −f(x).
- If f(−x) = f(x), the function is even. If f(−x) = −f(x), it is odd. Otherwise it is neither.
Example 1: f(x) = x4 − 3x2 + 1
f(−x) = (−x)4 − 3(−x)2 + 1 = x4 − 3x2 + 1 = f(x). The function is even.
Example 2: f(x) = x3 − 5x
f(−x) = −x3 + 5x = −(x3 − 5x) = −f(x). The function is odd.
Example 3: f(x) = x2 + x
f(−x) = x2 − x. This equals neither f(x) nor −f(x). For a numerical check, f(2) = 6 and f(−2) = 2, which are neither equal nor opposite. The function is neither even nor odd.
Example 4: f(x) = x/(x2 + 1)
f(−x) = −x/(x2 + 1) = −f(x). The function is odd.
How does symmetry look on a graph?
The graph of an even function is a mirror image about the y-axis: folding the paper along the y-axis puts the two halves on top of each other. The graph of an odd function has point symmetry about the origin: rotating it by 180° around (0, 0) leaves it unchanged. For an odd function defined at 0, the graph must pass through the origin, since f(0) = −f(0) forces f(0) = 0.
Even and odd functions table
| Function | f(−x) | Type |
|---|---|---|
| f(x) = x2 | x2 | Even |
| f(x) = |x| | |x| | Even |
| f(x) = cos x | cos x | Even |
| f(x) = x | −x | Odd |
| f(x) = x3 | −x3 | Odd |
| f(x) = 1/x | −1/x | Odd |
| f(x) = sin x | −sin x | Odd |
| f(x) = 2x | 2−x | Neither |
| f(x) = x + 1 | −x + 1 | Neither |
| f(x) = √x | undefined for x > 0 | Neither |
What happens when you add or multiply even and odd functions?
- Even + even = even, and odd + odd = odd.
- Even · even = even, and odd · odd = even.
- Even · odd = odd.
- Even + odd is usually neither. For example x2 + x is neither.
Every function with a symmetric domain can be written as the sum of an even and an odd function: f(x) = [f(x) + f(−x)]/2 + [f(x) − f(−x)]/2. For f(x) = 2x, the even part is the hyperbolic cosine-like expression (2x + 2−x)/2 and the odd part is (2x − 2−x)/2.
Where do the names come from?
The terms come from the parity of the exponents in power functions. They were used in the eighteenth century, and Leonhard Euler discussed even and odd functions in his analysis texts. The idea has become important in the study of trigonometric and Fourier series, where cosines are even and sines are odd.
Where is symmetry used today?
The integral of an odd function over a symmetric interval [−a, a] is 0, while for an even function it equals twice the integral over [0, a]. This often saves a long calculation. In physics, the potential energy of a spring, kx2/2, is an even function, while the force it produces is an odd function of displacement. In signal processing, even and odd parts of a signal are analysed separately.
Which mistakes are common?
Assuming every function is even or odd
Most functions are neither. Check f(−x) explicitly.
Ignoring the domain
A function defined only for x ≥ 0 cannot be even or odd.
Sign errors when substituting −x
Put −x in brackets everywhere: (−x)3 = −x3, but (−x)2 = x2.
Thinking a constant plus an odd function is odd
x3 + 1 is neither even nor odd. The graph is shifted away from the origin, so the point symmetry is lost.