 Hello and welcome to the session. In this session we will learn about exponential functions. Now any function of the form x is equal to a raise to power x where a is greater than 0 and a is not equal to 1 is called an exponential function with base a and the graph of the function has the equation y is equal to a raise to power x is the exponent that is the independent variable which is x is the exponent that means the domain which is the independent variable is the exponent. Now let us learn how to draw a graph for an exponential function. Now before we draw the graph we should keep in mind the following facts. Log x to the base a is defined for positive values always positive. Log 1 to the base a is equal to 0 say power 0 is equal to 1 and next then a is greater than 1. Log infinity to the base a is equal to infinity infinity is equal to infinity. Log 0 to the base a is equal to minus infinity infinity is equal to 1 over a raise to power infinity which is equal to 1 over infinity which is equal to 0. And next when 0 is less than a less than 1 that is a is lying between 0 and 1 then log infinity to the base a is equal to minus infinity infinity is equal to 1 over a power a raise to power infinity which is equal to infinity is greater than 1. Log 0 to the base a is equal to infinity power infinity is equal to 1 over a raise to power minus infinity which is equal to 0 as 1 over a is greater than 1. And next is whether never vanishes finite values. For example in this we have to draw a graph for the exponential function y is equal to 3 raise to power x. Now here a is equal to 3 x is an independent variable which is the exponent here called the exponential function. Now for the solution first of all let me draw the table y is equal to 3 raise to power x equals to 3 raise to power 1 which is equal to 3. So for x is equal to 1 y is equal to 3. Now for x is equal to 3 raise to power 2 which is 9. So for x is equal to 2 y is equal to 9 equal to 0. Y is equal to 3 raise to power 0 which is 1. So for x is equal to 0, y is equal to 1, 40 minus 1, y is equal to 3 is to power minus 1 which is 1 by 3, this is equal to 0.3 therefore, y is equal to 0.3, y is equal to 3 is to power minus 2 which is equal to 1 over 3 is to power 2 which is 1 over 9 which is equal to 0.1. So for x is equal to minus 2, y is equal to 0.1 which is equal to 1 over 3 is to power 3 which is equal to 1 by 37 which is equal to 0.03. Therefore, for x is equal to minus 3, y is equal to 0.03. Now, on the graph, let us plot the point 1, 3 on the graph. So, this is the point 1, 3 on the graph, plot the point 2, 9 on the graph. So, this is the point 2, 9 on the graph, plot the point 0, 1 on the graph, 0, 1 on the graph. Hence, we are getting the graph of the exponential function y is equal to 3 is to power x. Now, we can observe here that the curve does not pass through the original that is 1, y will be equal to 0 that is x is equal to 2 is equal to minus infinity. So, from this side, we can observe infinity, t is 0. Now, putting x is equal to 0 here, y is equal to 3 is to power 0 that is y is at the point 0, 1. Now, as 3 is to power x that is y is positive whether x is positive or negative or in other. Now, from this side, we can observe increases. Then y increases that is y is the increasing function of x infinity, y tends to infinity. So, by this example, we have discussed the case 1 when a for example will be 0 and 1 that is, plot a graph for the exponential function y is equal to 1, y is to power 1 by 3 is lying between 0 and 1. Now, for this, we will draw a table for the different values of x and y. Now, putting the various values of x, we can find the different values of y. Now, for x is equal to minus 2, y is equal to 1 by 3 is to power minus 1, y is equal to minus 2, which is equal to 3 is to power 2 which is equal to 9. So, for x is equal to minus 2, y is equal to 9. Now, for x is equal to minus 1, y is equal to 1 by 3 is to power minus 1, which is equal to 3. Now, for x is equal to 0, y is equal to 1 by 3 is to power 0, which is equal to 1. So, for x is equal to 0, y is equal to 1. Now, for x is equal to 1, y is equal to 1 by 3 is to power 1, which is equal to 1 by 3 which is equal to 0.3. So, for x is equal to 1, y is equal to 0.3. Now, for x is equal to 2, y is equal to 1 by 3 is to power 2, which is equal to 1 by 3 square, which is equal to 1 by 9 and this is equal to 0.1. So, for x is equal to 2, y is equal to 0.1. And now, we have the points minus 2, minus 2, 9 on the graph. Point minus 1, 3 on the graph. So, this is the point minus 1, 3 on the graph and all these points on the graph. We have plotted all these points on the graph. Now, by joining all these points, we are getting the graph for the exponential function, y is equal to 1 by 3 whole raised to power x. Now, where we can observe that the curve does not pass through the origin O, this when y is equal to 0, that is when x will be equal to infinity. Therefore, the curve needs the positive direction of x axis at infinity, that is infinity y tends to 0. That is, you can see from the side y tends to 0, that is by putting x is equal to 0 where we are getting y is equal to 1. So, it is cutting the line to the point 0, 1. This is positive, whether x is positive or negative or in other words, the curve lies only in the first and second programs, that is minus infinity, y to half, we have discussed the second case, where a is lying between 0 and 1. Now, we have learnt about exponential functions and this concludes our session. Hope you all have enjoyed this session.