 Hello and welcome to the session. In this session we will discuss graphs of piecewise defined functions including step functions and absolute value functions. First of all we are going to discuss piecewise function. A piecewise function is defined by more than one formula that is a formula for each interval. Now we shall discuss graph of a piecewise function. Now let us discuss it with the help of an example. Draw the graph of piecewise function f defined by f of x is equal to minus 2 if x is less than or equal to minus 3 and f of x is equal to 1 if x is greater than minus 3. Here we see that this function is defined for all real values of x so we say that its domain is r that is the set of real numbers. Now if we see the first statement of the function that is for x less than or equal to minus 3 y that is equal to f of x is equal to minus 2. It means graph of f is a horizontal line at y is equal to minus 2 and also x is less than or equal to minus 3 where less than or equal to sign means minus 3 is the included point so graph must show a dark circle or closed point at x is equal to minus 3. Now see the second statement of the given function that is for x is greater than minus 3 y which is equal to f of x is equal to 1 it means graph of f is a horizontal line at y is equal to 1 and also x is greater than minus 3 and greater than sign means minus 3 is not included so graph must show a hollow circle at x is equal to minus 3 this is the graph of the given piecewise function here you can see the function f of x or y can take only two values that is minus 2 and 1 so range of this function will be equal to the set containing elements minus 2 1. Now we are going to discuss absolute value function or modulus function the function defined by f of x is equal to absolute value of x that is equal to x when x is greater than or equal to 0 and is equal to minus x when x is less than 0 is called absolute value function its domain is the set of real numbers are now we are going to discuss graph of absolute value function or modulus function let us discuss it with the help of an example that is graph the function f of x is equal to modulus of x minus 3 now we know that domain of this function will be equal to r that is the set of real numbers now let us draw its graph for this we make table of values now here in this table we have randomly taken x values as minus 1 0 1 3 5 and 7 now we will find values of y for the corresponding values of x Now for x is equal to minus 1 we have y is equal to f of minus 1 that is equal to modulus of minus 1 minus 3 which is equal to modulus of minus 4 that is equal to 4 so for x is equal to minus 1 y is equal to 4 similarly for x is equal to 0 y is equal to f of 0 that is equal to modulus of 0 minus 3 that is equal to modulus of minus 3 which is equal to 3. So for x is equal to 0, y is equal to 3. Similarly we will find all the values of y for the corresponding values of x and here we get for x is equal to 1, y is equal to 2, for x is equal to 3, y is equal to 0, for x is equal to 5, y is equal to 2 and for x is equal to 7, y is equal to 4. Now we shall plot these points on the graph. We have plotted these points on this graph. Then we have joined these points to form this curve. Now this is the required graph of the function f of x is equal to modulus of x minus 3. From the graph we see that f of x or y takes values greater than or equal to 0. So its range is semi closed interval 0 to infinity. Now we are going to discuss step function. A function f of x is equal to greatest integer of x where f of x is equal to n that is an integer such that n is less than or equal to x is less than n plus 1 is called step function or greatest integer function. The symbol greatest integer of x denotes the greatest integer less than or equal to x its domain is set of real numbers and range is set of integers. Step function is a special type of function whose graph is a series of line segments rather than being a continuous line. It looks like a series of small steps. Let us discuss this an example. Draw graph of f of x is equal to greatest integer of x minus 2. We know that the greatest integer function assigns the greatest integer less than or equal to each real number in an interval. So now we make its table of values. Here in this table we have taken x values in the first row, greatest integer of x values in the second row and f of x or y values in the third row and we have randomly taken these x values and now for these values we will find the corresponding values of greatest integer of x and f of x. Now here for x is equal to 1 the value of greatest integer of x will also be equal to 1 and the value of f of x that is the value of f of 1 will be equal to 1 minus 2 that is equal to minus 1. So for x is equal to 1 y is equal to minus 1. Now for x is equal to 1.1 the value of greatest integer of x will be equal to 1 and the value of f of x that is the value of f of 1.1 will be equal to 1 minus 2 that is equal to minus 1. So for x is equal to 1.1 the value of y is equal to minus 1. Similarly we have found the values of greatest integer of x and y for the remaining values of x. Now we shall plot the values of x and values of y on the graph and we get the following graph for the function f of x is equal to greatest integer of x minus 2. Here we see that the graph of step function looks like series of small steps. Thus in this session we have discussed graphs of piecewise defined functions including step functions and absolute value functions. This completes our session. Hope you enjoyed this session.