Even and odd functions

When you reach for it

Symmetry about the vertical axis and the origin describes two mirror images a graph can have: a fold along the y-axis, where each point (a, b) has the partner (-a, b), and a half turn about the origin, where the partner is (-a, -b). A drawing can suggest either one, but it shows only a window, and only as exactly as a pencil line allows.

The rule settles it exactly, for every input at once. Reach for this test when you have a formula and want to know its symmetry: before sketching, so that you plot only the half with x \ge 0 and mirror the rest; when checking a sketch; and later, when symmetry shortens a calculation. The same idea tests equations that are not functions, such as a circle.

The procedure

Even and odd functions

The graph of f is symmetric about the y-axis when every point (a, f(a)) has its partner (-a, f(a)) on the graph too. The point of the graph above -a is (-a, f(-a)), so the partner is there exactly when f(-a) = f(a). Asking this for every a gives the definition. A function f is even when

The graph is symmetric about the origin when every point (a, f(a)) has the partner (-a, -f(a)), which means f(-a) = -f(a). A function f is odd when

Both definitions need a symmetric domain: whenever x is in it, so is -x, or f(-x) has no meaning.

So the test is a calculation. Find f(-x) by putting -x in for every x in the rule, as in evaluating at an expression, and simplify. If the result is f(x), the function is even. If it is -f(x), the opposite of every term, it is odd. Otherwise the function is neither, which is the usual case: unlike integers, functions need not be one or the other.

In reflecting a graph, y = f(-x) was the graph flipped across the y-axis and y = -f(-x) the graph turned a half turn. An even function is one the flip leaves unchanged, and an odd function one the half turn leaves unchanged, since -f(-x) = f(x) says the same as f(-x) = -f(x).

Why the names come from powers

For a whole number n, (-x)^n = (-1)^n x^n, and (-1)^n is 1 when n is even and -1 when n is odd:

So every even power x^2, x^4, x^6, \ldots is an even function and every odd power x, x^3, x^5, \ldots an odd function; a constant c is c\,x^0, an even power. That is where the names come from, and the squaring and cubing functions are the first examples. A polynomial built only from even powers is even, one built only from odd powers is odd, and one with terms of both kinds is neither: replacing x by -x flips the signs of the odd-power terms and keeps the others, so the result matches neither f(x) nor -f(x).

Past polynomials the names keep their meaning but lose the exponents: |x| and \displaystyle \sqrt{9 - x^2} are even, and the reciprocal function \displaystyle \frac{1}{x} is odd.

Showing that it is not

One input shows a function is not even: if f(-a) \ne f(a) for a single a, the rule f(-x) = f(x) fails. Likewise one input with f(-a) \ne -f(a) shows it is not odd. The other direction needs algebra: f(-2) = f(2) and f(-5) = f(5) say nothing about x = 3. To show a function is even or odd, simplify f(-x) for a general x.

Equations that are not functions

An equation in x and y, such as x^2 + y^2 = 25, may fail the vertical line test, so there is no f to evaluate. The same idea works on the equation itself: a point (a, b) is on the graph when it makes the equation true, so each test asks whether the partner point makes it true as well.

“Unchanged” means the new equation simplifies to the old one, perhaps after multiplying both sides by -1, so it has exactly the same solutions. When it does not, one point that satisfies the old equation and fails the new one proves it.

A graph symmetric about the x-axis is almost never a function: a point (a, b) with b \ne 0 and its partner (a, -b) stand on the same vertical line. So the x-axis test belongs to equations, not functions.

Sums and products

For functions f and g with the same symmetric domain:

The product column behaves like exponents: x^2 \cdot x^3 = x^5, and an even exponent plus an odd one is odd. Each entry takes one line. For two odd functions,

so the product is even. A quotient \displaystyle \frac{f}{g} follows the product column, since \displaystyle \frac{1}{g} has the same type as g. Example 7 runs the sum row backwards, splitting a function into an even part and an odd part.

Worked examples

Boundaries

The domain must be symmetric. \displaystyle \sqrt{x} and \displaystyle \frac{1}{x - 2} are neither, because some input has no opposite in the domain.

“For every x” is the whole condition. Agreement at a few inputs proves nothing; one disagreement disproves it.

Even and odd are neither the only choices nor opposites. Most functions are neither, and f(x) = 0 is both, the only function that is (Exercise 22).

The test sees only the axes and the origin. A graph symmetric about another line or point, such as y = (x - 1)^2 about x = 1, passes none of these tests. A shift left or right moves the mirror off the axis, and a shift up or down moves an odd function’s center off the origin, which is why combining a shift with other moves usually destroys evenness or oddness.

For functions, symmetry about the x-axis is ruled out unless every point is on the x-axis. Test it on equations.

Where this goes next

The powers that named these functions return in Degree, end behavior and the shape of a polynomial graph, where an even or odd leading power decides how the two ends of a polynomial graph behave. The half-turn symmetry of an odd function is the point symmetry of Rotational and point symmetry. In trigonometry, Unit circle symmetry and even-odd identities shows that cosine is even and sine is odd, the even-odd identities. In calculus, the properties of definite integrals include the fact that over an interval centered at 0 an odd function’s integral is 0, the two sides canceling, and an even function’s is twice the integral over the right half.

Exercises

Polynomials

Decide whether each function is even, odd or neither.

1. f(x) = x^4 + 7x^2 - 3
2. g(x) = 4x^3 - 9x
3. h(x) = x^3 + 1
4. q(x) = -x^7 + 3x^5 - x^3
5. r(x) = (x + 2)^2

Beyond polynomials

Decide whether each function is even, odd or neither.

6. \displaystyle f(x) = \frac{x^2}{x^4 + 1}
7. \displaystyle g(x) = \frac{x^3}{x^2 - 9}
8. \displaystyle h(x) = \frac{x}{x - 2}
9. \displaystyle k(x) = \sqrt{x^2 + 4}
10. \displaystyle m(x) = x\sqrt{x^2 + 1}

Equations

Test the graph of each equation for symmetry about the x-axis, the y-axis and the origin.

11. y = x^2 - 4
12. x = y^4 - 3
13. xy = 4
14. 4x^2 + 9y^2 = 36
15. y^3 = x
16. x^2 + y^2 - 6y = 0

Sums, products and parts

17. Without multiplying out, decide whether each is even, odd or neither: (a) x^2(x^3 - x), (b) (x^3 + x)^2, (c) \displaystyle \frac{x^5 - x}{x^2 + 3}.
18. f is odd and g is even, both defined for every real number. Decide whether each is even, odd or neither: (a) f(x)^3, (b) g(x) + 5, (c) f(x) + 5.
19. Write p(x) = x^3 + 2x^2 - 5x + 6 as the sum of an even function and an odd function.
20. Vlasta models the bend of another tuning fork as \displaystyle v(x) = \frac{x^4}{10} - \frac{x^3}{5} + \frac{x^2}{4} for -2 \le x \le 2, in centimeters. (a) Is the outline symmetric about the y-axis? (b) Find the heights v(2) and v(-2) of the two ends. (c) Find the even part of v and the height of its ends.

Reasoning

21. Explain why f(-2) = f(2) and f(-5) = f(5) do not show that f is even, while f(-1) \ne f(1) does show that f is not even. Give a function with the first two properties that is not even.
22. Prove that a function defined for every real number that is both even and odd must be f(x) = 0.
23. Prove that a graph with any two of the three symmetries (about the x-axis, the y-axis, the origin) has the third.