 Hi and welcome to the session. Today we will learn about composition of functions and invertible functions. First of all let us see what is composition of functions. Let f from a to b and g from b to c be two functions position the functions f denoted by g o f is defined a function g o f from a given by g o f of x is equal to g of f of x for all x belonging to a. Let us take one example. Here we are given the function f from r to r defined as f of x is equal to 8x q and the function g from r to r defined as g of x is equal to x to the power 1 by 3. We need to find f o g and g o f and we need to show that f o g is not equal to g o f. So first of all let us find g o f of x this will be equal to g of f of x equal to g of f of x is 8x q. So here we will get 8x q. Now g of 8x q will be equal to 8x q to the power 1 by 3 and this will be equal to 2x. Now let us find out f o g of x so this will be equal to f of g of x to f of g of x is x to the power 1 by 3. So f of x to the power 1 by 3 will be equal to 8 into x to the power 1 by 3 whole q. So this is equal to x is equal to 2x and f o g of x is equal to 8x. So from here we can say that f o g is not equal to g o x move on to invertible function. A function is defined invertible a function g from y to g o f is equal to identity function on set A is equal to identity function on set. The function is called the inverse of function f and is denoted by inverse. Also if the function f is invertible then the function f must be 1 1 also conversely the function f is 1 1 then the function f must be invertible. Let's take one example here we are given the function f from n to high defined as f of x is equal to 4x plus 5 where y is the same thing to n such that y is equal to 4x plus 5 for some x belonging to n to show that f is invertible. So for this consider an arbitrary element y of belongs to the set y that means x is equal to y minus 5 upon f of x so this will be equal to g of f of x is 4x plus 5 so this will be 4x plus 5 of y is equal to y minus 5 upon 4. So here we will get y minus 5 upon and this will be equal to 5 upon 4 is equal to the identity function on set n the function on set that the function f is invertible thus is equal to the function g. Thus in this session we have learnt composition of functions and invertible functions. With this we finish this session hope you must have understood all the concepts goodbye take care and have a nice day.