Higher Engineering Mathematics

200 GRAPHS

− 3 03 x

27

− 27

y y = x 3

(a)

− 3 π/2−π −π/2 0 π/2 π 3 π/2 2 π x

y y = sinx

1

− 1

(b)

Figure 19.26

(a)

− 10 123 x

y

20

10

y = ex

(b)

0 x

y

Figure 19.27

of odd functions arey=x^3 andy=sinxas shown
in Fig. 19.26.
Many functions are neither even nor odd, two such
examples being shown in Fig. 19.27.

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. 19.28(a) and

is symmetrical about the origin and is thus anodd

function(i.e. tan (−x)=−tanx).

(b) A graph off(x) is shown in Fig. 19.28(b) and

is symmetrical about thef(x) axis hence the

function is anevenone, (f(−x)=f(x)).

−π 0 x

y y = tan x

− 2 π −π 0 π 2 π x

f(x)

2

− 2

π 2 π

(a)

(b)

Figure 19.28

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)=x in the range −π to π and is

periodic of period 2π.