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23.3 Odd and even functions 729
f(t)
t
f(t)
1
(a)
f(t)
1
– p –4
p –4
t
0
–1
t
–1
(b)
Figure23.7
The function: f (t) = sint.
Figure23.8
Examples ofodd functions:
(a)f(t) =t; { 1 t>0
(b)f(t) =
−1 t<0.
Now it is easy to check that the first term on the r.h.s. is even and the second term is
odd, so that we have expressed f (t) as the sum of an even and an odd component as
required.
Example23.9 Showthattheproductoftwoevenfunctionsisitselfanevenfunction.Determinewhether
the product of two odd functions is even or odd. Is the product of an even function and
anoddfunctioneven orodd?
Solution If f (t) andg(t) are even then f (−t) = f (t) andg(−t) = g(t). LetP(t) = f (t)g(t) be
the product of f andg. Then
P(−t) = f (−t)g(−t)
by definition
= f (t)g(t) since f andgare even
=P(t)
Therefore,P(−t) = P(t) and so the product f (t)g(t) is itself an even function. On the
otherhand, if f (t)andg(t) areboth oddwefind
P(−t) = f(−t)g(−t)
= (−f(t))(−g(t))
= f(t)g(t)
=P(t)
so thatthe product f (t)g(t)iseven.