Higher Engineering Mathematics

202 GRAPHS

Figure 19.30

Figure 19.31

It is noted from the latter example, that not all func-
tions have an inverse. An inverse, however, can be
determined if the range is restricted.

Problem 5. Determine the inverse for each of

the following functions:

(a)f(x)=x− 1 (b)f(x)=x^2 −4(x>0)

(c)f(x)=x^2 + 1

(a) Ify=f(x), theny=x− 1
Transposing forxgivesx=y+ 1
Interchangingxandygivesy=x+ 1
Hence iff(x)=x−1, thenf−^1 (x)=x+ 1

(b) Ify=f(x), theny=x^2 −4(x>0)

Transposing forxgivesx=

√

y+ 4

Interchangingxandygivesy=

√

x+ 4

Hence iff(x)=x^2 −4(x>0) then

f−^1 (x)=

√

x+4ifx>− 4

(c) Ify=f(x), theny=x^2 + 1

Transposing forxgivesx=

√

y− 1

Interchangingxandygivesy=

√

x−1, which

has two values.

Hence there is no inverse of f(x)=x^2 + 1 ,

since the domain off(x) is not restricted.

Inverse trigonometric functions

Ify=sinx, thenxis the angle whose sine isy.

Inverse trigonometrical functions are denoted by

prefixing the function with ‘arc’ or, more com-

monly,−^1. Hence transposingy=sinxforxgives

x=sin−^1 y. Interchangingxandygives the inverse

y=sin−^1 x.

Similarly,y=cos−^1 x,y=tan−^1 x,y=sec−^1 x,

y=cosec−^1 xandy=cot−^1 xare all inverse trigono-

metric functions. The angle is always expressed in

radians.

Inverse trigonometric functions are periodic so

it is necessary to specify the smallest or principal

value of the angle. For sin−^1 x, tan−^1 x, cosec−^1 x

and cot−^1 x, the principal value is in the range

−

π

2

<y<

π

2

. For cos−^1 xand sec−^1 xthe principal
value is in the range 0<y<π.
Graphs of the six inverse trigonometric functions
are shown in Fig. 33.1, page 333.

Problem 6. Determine the principal values of

(a) arcsin 0.5 (b) arctan(−1)

(c) arccos

(

−

√

3

2

)

(d) arccosec(

√

2)

Using a calculator,

(a) arcsin 0. 5 ≡sin−^10. 5 = 30 ◦

=

π

6

rador0.5236 rad

(b) arctan(−1)≡tan−^1 (−1)=− 45 ◦

=−

π

4

rador−0.7854 rad