Understanding Engineering Mathematics

Examples include:

f(x)= 5 x^2 , 3 x^4 +x^2

i.e. polynomials with only even powers ofx.
Anodd functionchanges sign with its argument

f(−x)=−f(x)

Examples aref(x)= 3 x,2x^3 −x, i.e. polynomials with only odd powers ofx.
The trig function cosx, studied in Chapter 6 is even:

cos(−x)=cosx

On the other hand sinxis odd:

sin(−x)=−sinx

The graph of an even function is symmetric about they-axis, while the graph of an odd
function is unchanged under a rotation of 180°– see Figure 3.3.

y

0 x

y = x^4

Even

y

0 x

y = x^3

Odd

Figure 3.3Even and odd functions.

Anyfunctionf(x)for whichf(−x)exists can be expressed as the sum of an even and
an odd function:

f(x)=

f(x)+f(−x)

2

even

+

f(x)−f(−x)

2

odd

note the ‘something for nothing’ trick of adding zero in the form

0 =

f(−x)

2

−

f(−x)

2