 Hello and welcome to the session. In this session we will discuss about functions. Basically a function is a special type of a relation. So we say a relation F which goes from a set A to a set B is set to be a function if every element of set A has one and only one image in set B. If F is a function from A to B which is written in this form and we have that the ordered pair AB belongs to the function F then F of A is equal to B where this B is called the image of A under F and A is called pre-image of B under F. Here the set A is the domain of the function F and set B is the codomain of function F and the set of images is called the range of the function. Consider a set A equal to 1, 2, 3, 4 and a set B equal to 1, 4, 9, 16, 25 let F be equal to XY such that X belongs to A, Y belongs to B and Y is equal to X square. Now what we do is we give different values to X from the set A and we get the corresponding values of Y using Y equal to X square. So we have F is equal to 1, 1, 2, 4, 3, 9, 4, 16. Clearly F which goes from the set A to set B is a function F since every element in set A has a unique image in set B under F. Then domain of the function F is the set A that is equal to 1, 2, 3, 4 then codomain of the function F is the set which is 1, 4, 9, 16, 25. Then we have range of the function F is the set of images which is 1, 4, 9, 16. Now next we define a real valued function, a function which has either R or one of its subsets as its range is called a real valued function and further if its domain is also either R or a subset of R it is called a real function. Now we shall discuss some functions and their graphs. First let's consider identity function. Here we have let R be the set of real numbers then we define the real valued function F which goes from R to R by Y equal to F of X or equal to X for each X belongs to R such a function is called an identity function. Here the domain and range of F is R. This is the graph for the identity function which is a straight line as you can see passes through the origin. Next is the constant function we define a function F which goes from R to R by Y equal to F X equal to C for each X belongs to R and where we have the C is some constant. Now here domain of the function F is R that is the set of real numbers and range of F is the single term C. The graph for the constant function is the line parallel to the X axis like this is the graph for the function F X equal to 3 for each X belongs to R. Next we have polynomial function that is a function F which goes from R to R is said to be a polynomial function if for each X in R we have Y is equal to F X equal to A naught plus A 1 X plus A 2 X square plus and so on up to A n X to the power n where this n is non-negative integer and A naught A 1 A 2 and so on up to A n belongs to R. Consider a function F which goes from R to R defined by Y equal to F X equal to X square where each X belongs to R. This is the graph for the function Y equal to F X equal to X square. Now domain of this function F is equal to X such that X belongs to R and range of the function F is equal to the set X such that X is greater than equal to 0 for each X belongs to R. Next we shall discuss rational functions. These are the functions of the type F X upon G X where we have F X and G X are the polynomial functions of X defined in a domain and where we have G X is not equal to 0. Let's define a function F which goes from R minus 0 to R defined by F X equal to 1 upon X where we have X belongs to R minus 0. This is the graph for this function F equal to 1 upon X and we have that the domain of the function F is the all real numbers except 0 then its range is also all real numbers except 0. Then we have modulus function that is the function F goes from R to R defined by F X equal to modulus X for each X belongs to R this is called the modulus function and we have that for each non-negative value of X F X is equal to X that is we have F X is equal to X when X is greater than equal to 0 and for negative values of X that is when X is less than 0 the value of F X is negative of the value of X that is minus X and this is the graph for the modulus function F X equal to modulus X. The next is the signum function that is the function F which goes from R to R defined by F X equal to 1 when we have X is greater than 0 and F X equal to 0 when X is equal to 0 and F X equal to minus 1 when X is less than 0 this is the signum function. This is the graph for the signum function where we have F X is equal to modulus X upon X and X is not equal to 0 and we have F X is equal to 0 for X equal to 0 then domain of the function F that is the signum function is R and range of the function F is the set minus 1 0 1. Now next is the greatest integer function that is the function F which goes from R to R defined by F X equal to the greatest integer function X where X belongs to R this assumes the value of the greatest integer less than or equal to X such a function is called the greatest integer function and from the definition of the greatest integer function we have that greatest integer function X is equal to minus 1 when we have X is greater than equal to minus 1 and less than 0 then value of this is equal to 0 when we have X is greater than equal to 0 and less than 1 then value of the greatest integer function X is 1 when X is greater than equal to 1 and less than 2 and so on. This is the graph for the greatest integer function X. Next we shall discuss algebra of real functions. First we have addition of two real functions that is we have a function F which goes from X to R and a function G which goes from X to R. These are the two real functions where we have X is a subset of R then we define F plus G as a function which goes from X to R by F plus G of X is equal to F X plus G X for all X belongs to X then we have subtraction of real function from another consider F which goes from X to R real function and G which goes from X to R be another real function where again we have X is a subset of R then we define F minus G as a function which goes from X to R by F minus G of X is equal to F X minus G X for all X belongs to X. Next is multiplication by a scalar we consider a real valued function F which goes from X to R and we have alpha be any scalar then the product of the function F and the scalar is alpha F it is a function defined from X to R such that alpha F of X is equal to alpha into F X where we have X belongs to X then we have multiplication of two real functions we consider a real function F which goes from X to R and a real function G which again goes from X to R then the product is defined by F G which goes from X to R such that we have F G of X is equal to F X multiplied by G X for all X belongs to X this is also called point wise multiplication then we have question of two real functions consider a real function F which goes from X to R and another real function G which goes from X to R where we have X is a subset of R then the question of F by G denoted by F upon G is defined by F by G of X is equal to F X upon G X for all X belongs to X and where we have G X is not equal to zero let the function F X be equal to X square this is a real function then a function G X equal to three X plus two which is again a real function now F plus G of X is equal to F X that is X square plus G X that is three X plus two then F minus G of X is equal to F X that is X square minus G X that is three X plus two which is finally equal to X square minus three X minus two then F G of X is equal to F X which is X square multiplied by G X that is three X plus two now this is equal to three X cube plus two X square now F by G of X is equal to F X that is X square upon G X that is three X plus two provided that X is not equal to minus two upon three this completes the session hope you have understood the concept of functions