Chapter 6 Transcendental Functions and Their
Inverse
6.1 Inverse Functions
Definition 1:
f and g are inverse functions if and only if

f g x   x

g f x  x
Definition 2:
The function y  f x  is one-to-one, if x1 , x2  a, b and x1  x2 ,
then f x1   f x2  .
Theorem 1:
If f is either an increasing or a decreasing function on a, b, then f
is one-to-one on a, b.
Theorem 2:
If f is continuous on a, b and has a derivative on
a, b
satisfying
f ' x   0 for every x in a, b , then f is one-to-one on a, b.
[justifications of theorem 2:]
Suppose f is not one-to-one. Then, there exists x1  x2 such that
f x1   f x2  . Without loss of generality, let x1  x2 . By mean-value
1
theorem, there exists c  x1 , x2  such that
0  f x1   f x2   f ' c x2  x1  .
'
Since x1  x2  0 , f c   0 . This leads to a contradictory. Therefore,
f is one-to-one.
Theorem 3:
Let f be continuous on a, b and let c  f a  and d  f b . If
f has an inverse g  f 1 , then g is continuous on c, d .
Theorem 4:
y  f x  is differentiable and one-to-one in a, b . Let x  a, b
'
and f x   0 . Let g be inverse function of f and g f x   x .
Then, g is differentiable at x and
g '  f  x  
1
f ' x 

dx
1

.
dy  dy 
 dx 


[intuition of theorem 4:]
1
dx dg  f  x 

 g '  f  x  f '  x by chain rule 
dx
dx
Thus,
g '  f  x  
1
f ' x 

dx
1

.
dy  dy 
 dx 


[justifications of theorem4:]
y  f x, y  f x  x  f x  f x  x  y  y  y  f x  x .
2
Since y  f x  is continuous, x  0  y  0 . Then,
g  y  y   g  y  g  f x  x   g  f x 
x  x  x


y
y
f x  x   f x 
x
1


f x  x   f x  f x  x   f x 
x
Thus,




 g  y  y   g  y  
1
g '  f  x   g '  y   lim 

lim

 x0 f x  x   f x  
y 0
y




x


1

 f x  x   f x 
lim 

x 0
x

1
 '
f x 
Note:
The graphs of f and f
1
are reflections of one another about the
line y  x .
3