Higher Engineering Mathematics

FUNCTIONS AND THEIR CURVES 201

C

(a) A graph ofy=lnxis shown in Fig. 19.29(a)
and the curve is neither symmetrical about the
y-axis nor symmetrical about the origin and is
thusneither even nor odd.

(b) A graph ofy=xin the range−πtoπis shown
in Fig. 19.29(b) and is symmetrical about the
origin and is thus anodd function.

12 3 4 x

y = lnx

y

1.0

0.5

−0.5

− 2 π −π^0 π^2 π x

−π

π

y y = x

(a)

(b)

0

Figure 19.29

Now try the following exercise.

Exercise 86 Further problems on even and

odd functions

In Problems 1 and 2 determine whether the given

functions are even, odd or neither even nor odd.

1. (a)x^4 (b) tan 3x (c) 2e^3 t (d) sin^2 x
   [
   (a) even (b) odd
   (c) neither (d) even

]

2. (a) 5t^3 (b) ex+e−x (c)

cosθ

θ

(d) ex

[

(a) odd (b) even

(c) odd (d) neither

]

3. State whether the following functions, which
   are periodic of period 2π, are even or odd:

(a)f(θ)=

{

θ, when−π≤θ≤ 0

−θ, when 0≤θ≤π

(b)f(x)=

⎧

⎨

⎩

x, when−

π

2

≤x≤

π

2

0, when

π

2

≤x≤

3 π

2

[(a) even (b) odd]

19.6 Inverse functions

Ifyis a function ofx, the graph ofyagainstxcan be

used to findxwhen any value ofyis given. Thus the

graph also expresses thatxis a function ofy.Two

such functions are calledinverse functions.

In general, given a functiony=f(x), its inverse

may be obtained by interchanging the roles ofxand

yand then transposing fory. The inverse function is

denoted byy=f−^1 (x).

For example, ify= 2 x+1, the inverse is obtained

by

(i) transposing forx, i.e.x=

y− 1

2

=

y

2

−

1

2

and

(ii) interchangingxandy, giving the inverse as

y=

x

2

−

1

2

Thus iff(x)= 2 x+1, thenf−^1 (x)=

x

2

−

1

2

A graph off(x)= 2 x+1 and its inversef−^1 (x)=

x

2

−

1

2

is shown in Fig. 19.30 andf−^1 (x) is seen to

be a reflection off(x) in the liney=x.

Similarly, ify=x^2 , the inverse is obtained by

(i) transposing forx, i.e.x=±

√

yand

(ii) interchanging x and y, giving the inverse

y=±

√

x.

Hence the inverse has two values for every value

ofx. Thusf(x)=x^2 does not have a single inverse. In

such a case the domain of the original function may

be restricted toy=x^2 forx>0. Thus the inverse is

theny=+

√

x. A graph off(x)=x^2 and its inverse

f−^1 (x)=

√

xforx>0 is shown in Fig. 19.31 and,

again,f−^1 (x) is seen to be a reflection off(x)in

the liney=x.