Page 119 - DMTH401_REAL ANALYSIS

Unit 9: Functions

Notes
 
f : ] – , [  R defined by
2 2
 
f(x) = tan x, – , < x <
2 2
is one-one and onto. Hence it has an inverse. The inverse function is called the inverse tangent
of x and is denoted by tan –1 x (or by arctan x). In other words,

y = tan x  x = tan y,
–1
 
where – < y < and –  < x < + .
2 2
Thus, we have the following definition:
Definition 13: Inverse Tangent Function

 
A function g : R  ], – , [ defined by
2 2
g(x) = tan x, " x R
–1
is called the inverse tangent function.

Task Define the inverse cotangent, inverse secant and inverse cosecant function. Specify
their domain and range.

Now, before we proceed to define the logarithmic and exponential functions, we need the
concept of the monotonic functions. We discuss these functions as follows:

9.3.1 Monotonic Functions

Consider the following functions:

(i) f(x) = x, " x  R.
(ii) f(x) = sin x, " x [–/2, /2].
(iii) f(x) = –x , " x [0, [,
2
(iv) f(x) = cos x, " x [0, ].
Out of these functions, (i) and (ii) are such that for any x,, x in their domains,
2
x, < x  f(X )  f(x ),
2 1 2
whereas (iii) and (iv) are such that for any x,, x in their domains,
2
x, < x  f(X )  f(x ).
2 1 2
The functions given in (i) and (ii) are called monotonically increasing while those of (iii) and (iv)
are called monotonically decreasing. We define these functions as follows:
Let f : S  R (S  R) be a function

(i) It is said to be a monotonically increasing function on S if
x < x,  f(X ) < f(X ) for any x,, x S
1 1 1 2

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