188 Higher Engineering Mathematics
(a) A graph ofy=lnx is shown in Fig. 18.29(a)
and the curve is neither symmetrical about the
y-axis nor symmetrical about the originand is thus
neither even nor odd.
(b) A graph ofy=xin the range−πtoπis shown in
Fig. 18.29(b) and is symmetrical about the origin
and is thus anodd function.
(a)
y
x
yIn x
0.5
1.0
0.5
02341
(b)
y
x
yx
0
2 2
Figure 18.29
Now try the following exercise
Exercise 78 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.
- (a)x^4 (b) tan3x (c) 2e^3 t (d) sin^2 x
[
(a)even (b)odd
(c)neither (d)even
]
- (a) 5t^3 (b) ex+e−x (c)
cosθ
θ
(d) ex
[
(a)odd (b)even
(c)odd (d)neither
]
- 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]
18.6 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.Twosuch
functions are calledinverse functions.
In general, given a functiony=f(x), its inverse may
be obtained by interchanging the roles ofxandyand
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) interchanging x and y, giving the inverse as
y=
x
2
−
1
2
Thus iff(x)= 2 x+1, thenf−^1 (x)=
x
2
−
1
2
A graph of f(x)= 2 x+1 and its inversef−^1 (x)=
x
2
−
1
2
is shown in Fig. 18.30 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.
Thus f(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.Agraphoff(x)=x^2 and its inverse