INTERVALS

CLOSED INTERVAL:

Let a and b be two given real numbers such that a < b. Then the set of all real numbers x such that a ≤ x ≤ b is called a closed interval and is denoted by [a, b].

i.e., [a, b] = {x ϵ R| a ≤ x ≤ b}

OPEN INTERVAL:

Let a and b be two given real numbers such that a < b. Then the set of all real numbers x such that a ≤ x ≤ b is called a closed interval and is denoted by (a, b).

i.e., (a, b) = {x ϵ R| a ≤ x ≤ b}

SEMI-CLOSED OR SEMI OPEN INTERVAL:

If a, b are two given real numbers such that a < b then the sets (a, b) = {x ϵ R| a ≤ x ≤ b} and (a, b) = {x ϵ R| a ≤ x }   are known as semi-closed or semi-open intervals and are also denoted by ] a, b[ and [a, b] respectively.

REAL FUNCTIONS:

REAL FUNCTION

If the domain and co-domain of a function are subsets of R (set of all real numbers). It is called a real valued function or in short a real function.

DESCRIPTION OF AREAL FUNCTION

If f is a real valued function with finite domain, then f can be described by listing the values which it attains at different points of its domain. However, if the domain of a real function is an infinite set, then, f cannot be described by listing the values at points in its domain. In such cases real functions are generally described by some general formula or rule like f (x) = x² + 1 or f (x) = 2sinx + 3 etc.

EXAMPLE:

If  prove that 

SOLUTION:

We have,

DOMAIN AND RANGE OF A REAL FUNCTION:

DOMAIN:

Generally real functions in calculus are described by some formula and their domains are not explicitly stated. In such cases to find the domain of a function f (say) we use the fact that the domain is the set of all real numbers x for which f (x) is a real number.

RANGE:

As discussed, the range of a function f (x) is the set of values of f (x) which it attains at points in its domain. For a real function the co-domain is always a subset of R. So, range of a real function f is the set of all points  such that y = f (x) where x ϵ Dom f (x)

ALGORITHM

STEP 1:

Put f (x) = y

STEP 2:

Solve the equations in step 1 for x to obtain x = ϕ (y).

STEP 3:

Find the values of y for which the values of x, obtained from x = ϕ (y) are in the domain of f.

STEP 4:

The set of values of y obtained in step 3 is the range of f.

EXAMPLE:

Find the domain and range of the function 

SOLUTION:

We have, 

Domain: We have,

-1 ≤ cos 3x ≤ 1 for all x ϵ R,

=> -1 ≤ -cos3x ≤ 1

=> -1 ≤ 2 -cos 3x ≤ 3 for all x ϵ R,

=> f (x) is defined for all x ϵ R

So, domain (f) = R.

Range: Let f (x) = y Then

So, range of .