Section 1-4-4

Chapter 1
Relationships

1.4.4 Symmetries: Odd and Even Functions

Other important functional equations describe symmetries — of functions that happen to have symmetries.

Definitions A function has even symmetry (or is an even function) if it satisfies the functional equation

f (- a) = f (a) .

Similarly, a function has odd symmetry (or is an odd function) if it satisfies

f (- a) = - f (a) .

Example 3

What is the symmetry of the squaring function, $f (x) = x^{2} ?$ (See Figure 2.)

Figure 2 The graph of

y = x^{2}

Solution This function has even symmetry, because

f (- a) = {(- a)}^{2} = a^{2} = f (a)

Example 4

What is the symmetry of the multiply-by-5 function, $f (x) = 5 x ?$ (See Figure 3.)

Figure 3 The graph of

y = 5 x

Solution This function has odd symmetry:

f (- a) = 5 (- a) = - 5 a = - f (a) .

Power functions with even and odd powers provide additional simple examples — as well as a reason for the names "even" and "odd."

Example 5

We saw in Example 3 that the squaring function is even. In a similar manner, if

f (x) = x^{3},

then

f (- a) = {(- a)}^{3} = - a^{3} = - f (a),

so the cubing function is odd (see Figure 4).

Figure 4 The graph of $y = x^{3}$

Activity 4

In Figure 5, we have sketched half the graph of a function, the half for positive values of the independent variable.

Figure 5 Half of the graph of a function

Assume the function is even, and, on a piece of scratch paper, sketch the whole graph. For a typical point $(a, b)$ on the graph, we have indicated the locations of the points $(- a, b),$ $(a, - b),$ and $(- a, - b)$ . Think about which of these points must lie on the graph of an even function. Apply the same reasoning to all the other points on the graph.
Now assume the function is odd, and sketch the whole graph.