Sequences & Functions

Functions
In the study of natural phenomena and the solution of technical and mathematical problems, we find it

necessary to consider the variation of one quantity as dependent on the movement of another.

For example expenditure is directly or indirectly related with the income or the price of goods prevalent in the market. Also we can say a distance covered by a moving particle is dependent upon the speed of the vehicle. Hence we can say the expenditure or the path traversed is a function of the price/demand or the time. In other words word if we say that the area of a circle 'A' depends upon the size of the radius 'r' here we mean that the area is a function of 'r'

Definition: If to each value of a variable 'x' (within a certain range) there corresponds one definite value of another variable 'y', then 'y' is called the function of 'x' ory = f(x)
According to modern concept, a function is a relation between two sets A and B in which each element of set A is matched to one and only one element of set B (such a relation is called single-valued and everywhere defined). In other words it reads 'f' is a function of A into B.
In a function each element of A must be matched to one and only one element of B.

Types of functions
Power function
General exponential function
Logarithmic function
Trigonometric functions
The rational integral function
Inverse function
Constant function
Composite function
Inverse Function

f -1(x) is known as Inverse function.
e.g. If f(x) = x2 then y(say) = x2, x2 = y, x =
? f -1(x) =
f -1(16). = ± 4,
f -1(9). = ± 3,
f -1(1). = ± 1,
f -1(25). = ± 5,

Constant Function
If f(x) =12, then f(a) = 12, f(0) = 12, f(5) = 12
All these are examples of Constant Function where the value of f(x) = y does not change with a change in the value of 'x'.

Composite Function (g.f.) is called a composite of 'f' and 'g'. 'g .f ' is real 'g circle f' OR 'g operation f' OR 'g composite f' are other terminologies for(g . f.). [Note that function f is applied first, and then g] (g .f) (a) = g[f(a)]


Domain and Range of a function: The set of values of 'x' for which the values of the function 'y' are determined by the rule f(x) is called the domain of function. The set of values of 'y' determined at various possible values of 'x' is called the range of the function.

A function is summarily a mapping or transformation of 'x' into 'y' or f(x). The variable 'x' presents elements of the domain and is called the independent variable. The variable 'y' representing the elements of the range is called the dependent variable.

Value of a Function at a Point
f(a) is the value of the function f(x) when x takes the value 'a' i.e., when 'x' is replaced by 'a'.