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23.3 Odd and even functions 727
f(t)
1
–2p
–p
– p –4 p –4
p
2p
t
Figure23.5
Thefunction: f (t) = cost.
f(t)
f(t)
3
(a)
Figure23.6
Examples ofeven functions:(a) f (t) =t 2 ;(b) f (t) = 3.
t
(b)
t
More generally, any function with the property that f (t) = f (−t) for any value of
its argument,t, issaidtobe aneven function.
If f (t) = f (−t) then f isaneven function.
In particular, the set of functions cosnωt, for any integern, is even. The graphs of all
even functions are symmetrical about the vertical axis -- or equivalently, the graph on
the left of the origin can be obtained by reflecting in the vertical axis that on the right.
Someother examples ofeven functions areshown inFigure 23.6.
Sketchingagraphshowsuptherequiredsymmetryimmediately.However,evenfunctions
can be identified by an algebraic approach as shown in Example 23.6. Given any
function f (t), weexamine f (−t)tosee if f (t) = f (−t).
Example23.6 Show that f (t) =t 2 is even.
Solution We can argue asfollows. If
f(t)=t 2
then
f(−t) = (−t) 2
=t 2
=f(t)
so that f (t)iseven, by definition.