What is an Odd Function? Definition and Examples

What is an odd function? A function f is an odd function if f(-x) = -f(x) for all x in the domain of f.

For example f(x) = x3 is an odd function

f(-2) = (-2)3 = -2 × -2 × -2 = -8 and f(2) = (2)=  2 × 2 × 2 = 8

f(-2) = -8 = -f(2)

f(-1) = (-1)3 = -1 × -1 × -1 = -1 and f(1) = (1)=  1 × 1 × 1 = 1

f(-1) = -1 = -f(1)

In general, let x represent any number in the domain of f.

Then, f(-x) = (-x)3 = -x × -x × -x = -x × x2 = -x3 and f(x) = x3

f(-x) = -x3 = -f(x)

For f(2) = 8 and f(-2) = -8, the points are (2, 8) and (-2, -8)

For f(1) = 1 and f(-1) = -1, the points are (1, 1) and (-1, -1)

Notice that f(0) = 03 = 0, so there is a single point (0,0)

Now, try to graph these five points on a coordinate system.

(2, 8), (-2, -8), (1, 1), (-1, -1), and (0, 0)

From the graph above, we can make the following two important observations:

• If a point (-x,-y) is on the graph, the point (x,y) is also on the graph.
• The function is symmetric with respect to the origin.

More examples of odd functions

• f(x) = x3 - 8x
• f(x) = x5
• f(x) = sin(x)
• f(x) = x7

f(x) = x3 + 1 is not an odd function.

f(-x) = -x3 + 1

f(-x) = -(x3 - 1)

Since x3 - 1 ≠ x3 + 1, f(-x) ≠ -f(x)