Functions and their curves 187
Odd functions
A functiony=f(x)is said to beodd iff(−x)=−f(x)
for all values of x. Graphs of odd functions are
always symmetrical about the origin. Two examples
of odd functions arey=x^3 andy=sinxas shown in
Fig. 18.26.
Many functions are neither even nor odd, two such
examples being shown in Fig. 18.27.
(a)23y(^3) x
27
227
y 5 x^3
0
1
0
(b)
y
x
21
2
y 5 sinx
23
2
3
2
2 ^2
2
2
Figure 18.26
(a)
y
(^1123) x
20
10
0
yex
y
(^0) x
(b)
Figure 18.27
Problem 3. Sketch the following functions and
state whether they are even or odd functions:
(a)y=tanx
(b)f(x)=
⎧
⎪⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎪⎩
2 , when 0≤x≤
π
2
− 2 , when
π
2
≤x≤
3 π
2
,
2 , when
3 π
2
≤x≤ 2 π
and is periodic of period 2π.
(a) A graph ofy=tanxis shown in Fig. 18.28(a) and
is symmetrical about the origin and is thus anodd
function(i.e. tan(−x)=−tanx).
(b) A graph of f(x)is shown in Fig. 18.28(b) and
is symmetrical about the f(x)axis hence the
function is anevenone,(f(−x)=f(x)).
(a)
y
0 2 x
ytanx
(b)
f(x)
0 2 x
2
2
2
Figure 18.28
Problem 4. Sketch the following graphs and state
whether the functions are even, odd or neither even
nor odd:
(a) y=lnx
(b) f(x)=xin the range−πtoπand is
periodic of period 2π.