Functions Limits and Continuity

 

Functions

f is a function from set A to a set B if each element x in A can be associated with a unique element in B.

The unique element B which f associates with x in A denoted by f (x).

Domain

In the above definition of the function, set A is called domain.

Co-Domain

In the above definition of the function, set B is called co-domain.

Real Function

A real valued function f : A to B or simply a real function 'f ' is a rule which associates to each possible real number xA, a unique real number f(x)B, when A and B are subsets of R, the set of real numbers.

In other words, functions whose domain and co-domain are subsets of R, the set of real numbers, are called real valued functions.

Value of a Function

If 'f ' is a function and x is an element in the domain of f, then image

f(x) of x under f is called the value of 'f ' at x.

Types of Function and their Graphs

Constant function

A function f : A ® B Such that A, B Ì R, is said to be a constant function if there exist K Î B such that f(x) = k.

Domain = A

Range = {k}

The graph of this function is a line or line segment parallel to x-axis. Note that, if k>0, the graph B is above X-axis. If k<0, the graph is below the x-axis. If k = 0, the graph is x-axis itself.

Identity function

A function f : R® R is said to be an identity function if for all x Î R, f(x) = x.

Domain = R

Range = R

Polynomial function

A function f : R® R is said to be a polynomial function if for each x Î R, f(x) is a polynomial in x.

f(x) = x³ + x² + x

Modulus function

f : R ® R such that f(x) = |x|,  is called the modulus function or absolute value function.

Domain = R

Square root function

Since square root of a negative number is not real, we define a function f : R⁺ ® R such that

Greatest integer function or Step function (Floor function)

f (x) = [x] = greatest integer less than or equal to x

[x] = n, where n is an integer such that

Smallest integer function (Ceiling function)

For a real number x, we denote by [x], the smallest integer greater than or equal to x. For example, [5 . 2] = 6, [-5 . 2] = -5, etc. The function f:RR defined by

f(x) = [x], xR

is called the smallest integer function or the ceiling function.

Domain: R

Range : Z

Exponential function

The exponential function is defined as f(x) = e^x. Its graph is

Logarithmic function

Logarithmic function is f (x) = log x. Its graph is

Trigonometric functions

Trigonometric functions are sinx, cosx, tanx, etc. The graph of these functions have been done in class XI.

Inverse functions

Inverse functions are sin^-1x, cos^-1x, tan^-1x etc. The graph of these functions have been done in class XI.

Signum functions

Odd function

A function f : AB is said to be an odd function if

f(x) = - f(-x) for all xA

The domain and range of f depends on the definition of the function.

Examples of odd function are

y = sinx, y = x³, y = tanx

Even function

A function f : AB is said to be an even function if

f(x) = f(-x) for all xA.

The domain and range of f depends on the definition of the function.

Examples of even function are

y = cosx, y = x², y = secx

A polynomial with only even powers of x is an even function.

Reciprocal function