Functions
Print this page
Email this Page
FUNCTIONS
Definition
A function is a rule that assigns to each element in a set X, one and only one element of a set Y. In general, the sets X and Y need not to be sets of real numbers. However, we consider only those functions for which X and Y are both subsets of the set of real numbers.
A function f from a set X to a set Y is defined as follows:
f: X èY ( read as function from X to Y) is said to be a function if it satisfies the following two conditions
-
For every x E X, there exist y E Y such that y = f(x)
-
For every x E X, y = f(x) is unique. In other words, for all x1, x2E X, x1 = x2=> f(x1) = f(x2) The set X is called the domain of the function f.
NOTE: The set X in the above definition is called the domain of the function, and the set Y is called the co domain of the function Y. For any x E X, the element y = f(x) in Y is called the image of x and x is called the preimage of y. The set of images of all elements of X is called the range of the function f. It is denoted by f(X) and is a subset of the co-domain Y.
NOTE: If x is an element in the domain of any function f, then the definition of this function requires that f assign one and only one value to x. This implies that a function cannot be multiple-valued. Thus, the expression ±(x)1/2 is not a function, since it assigns two values to each positive x.
The Range of a function is defined as
f(X) = {y E Y : y = f(x) for some x E X}
The Domain and Range of some elementary functions are given below .
Function Domain Range
1/x R - {0} R - {0}
1/(x-a) R - {a} R - {0}
+2x [0, P) [0, P)
log x (0, P) R
sin x R [-1, 1]
cos x R [-1, 1]
tan x R-{(2n+1)p/2} R
cot x R - {np} R
sin-1x [-1,1] [-p/2, p/2]
cos-1x [-1, 1] [0, p]
tan-1x R (-p/2, p/2)
cot-1x R (0, p)
NOTE:
CONSTANT FUNCTIONS: A function that assigns the same value to every member of its domain is called a constant function. The domain of the constant function f(x)=c is R and its range is {c}
POLYNOMIAL FUNCTIONS
A function of the form cxn, where c is a constant and n is a non-negative integer, is called a monomial in x. Functions such as 5x3, -6x, 4x2, x12 etc. are monomials. A function that is expressible as the sum of finitely many monomials in x is called a polynomial in x. A function
F(x) = Anxn + An-1xn-1 + ………..+ A1x + a0
is a polynomial. The domain of a polynomial is R and its domain is a subset of R.