 Hello and welcome to the session. In this session we will discuss about the increasing and decreasing functions. We can analyze the function by its monotonicity. A function is said to be monotonic on an interval i if it is either increasing or decreasing in the interval i. Now we shall learn to analyze increasing and decreasing function graphically. Now let us look at the graph of the function y is equal to x square. We can determine whether the function is increasing or decreasing by viewing its graph from left to right. See the left side of the curve. Here the graph is decreasing i.e. the curve is moving downward till point with coordinates 00. From this origin i.e. point 00 the curve starts increasing i.e. it starts moving upward. Thus the graph of function f of x is equal to x square shows that the function is decreasing for x is less than 0 and the function is increasing for x greater than 0. Or we can say that it is decreasing for x belongs to the open interval minus infinity to 0 and increasing for x belongs to the open interval 0 to infinity. Now we are going to discuss how to analyze increasing and decreasing functions algebraically. A function f is increasing on an interval i if and only if for every a and b contained in interval i f of a is less than f of b for a is less than b. For example if we want to check whether a function f of x is equal to x square is increasing or decreasing on an interval i given by open interval 0 to infinity. Now here let us choose a, b belong to open interval 0 infinity such that a is less than b. Now let us take a is equal to 2 and b is equal to 5. Clearly here a is less than b and they both belong to the open interval 0 to infinity. Now here f of x is equal to x square. Now let us find f of a which will be equal to f of 2 and that is equal to 2 square which is equal to 4. Also f of b will be equal to f of 5 that is equal to 5 square which is equal to 25. Now here f of a is equal to 4 and f of b is equal to 25. We see that 4 is less than 25 which implies that f of a is less than f of b. So the function f of x is equal to x square is increasing on open interval 0 to infinity. Now a function is decreasing on an interval i if and only if for every a and b contained in interval i f of a is greater than f of b for a is less than b. Now if we again take the same example where the function given to us is f of x is equal to x square and we want to check whether f of x is increasing or decreasing on an interval i given by the open interval minus infinity to 0. Let us choose a b belong to open interval minus infinity to 0 such that a is less than b. Now let us take a is equal to minus 2 and b is equal to minus 1. Clearly here a is less than b and they both belong to the open interval minus infinity to 0. Now let us find f of a and this will be equal to f of minus 2 and this is equal to minus 2 whole square which is equal to 4. Also f of b will be equal to f of minus 1 and this is equal to minus 1 whole square that is equal to 1. Now here f of a is equal to 4 and f of b is equal to 1. Here we see that 4 is greater than 1 which implies that f of a is greater than f of b for a less than b. So we can say that the function f of x is equal to x square is decreasing on interval that is on open interval minus infinity to 0. Also a function f remains constant on an interval i if and only if for every a and b contained in i f of a is equal to f of b for a is less than b. For example if we have the function f of x which is equal to minus x for x less than 0 and 1 by 2 for x greater than or equal to 0. Now let us check monotonicity of the function on semi closed interval from 0 to infinity. Now let us choose a, b which belong to the semi closed interval 0 to infinity such that a is less than b. So here let us take a is equal to 1 and b is equal to 2. Clearly here a is less than b and they both belong to the semi closed interval 0 to infinity. Now let us find f of a which will be equal to f of 1. Now here 1 is greater than or equal to 0 so we have the value of f of x as 1 by 2. So here f of 1 will be equal to 1 by 2 also f of b will be equal to f of 2 and since 2 is greater than or equal to 0 and for x greater than or equal to 0 the value of f of x is equal to 1 by 2. So here f of 2 will be equal to 1 by 2. Now f of a is equal to 1 by 2 and f of b is equal to 1 by 2. So here we see that f of 1 is equal to f of 2. So we can say that f of a is equal to f of b for a is less than b. Thus we can say that f of x is a constant function on semi closed interval 0 to infinity. Thus in this session we have learnt about the increasing and decreasing functions and we have also discussed how to analyse the function by its monotonicity. This completes our session. Hope you enjoyed this session.