 Hello and welcome to the session. In this session we will discuss domain and range of exponential functions. An exponential function is of the form f of x is equal to a into b raised to the power x, where a and b are real numbers and a is not equal to 0, b is greater than 0 and b is not equal to 1. Now let us consider the graph of the function f of x is equal to 2 raised to the power x. Let us see its domain and range graphically. Domain of a function are the possible x values and range of a function are the possible y values. Now if we look at this graph we see that the branch of the curve is existing for both negative and positive values of x. So domain of the function f of x is equal to 2 raised to the power x is all real numbers that is open interval from minus infinity to infinity. Now range are y values. Now if we look at this curve we see that it lies only in first and second quadrant that is the graph never intersects or lies below x axis. So the value of y is neither 0 nor negative. The curve is moving towards positive values of y. So range of the function f of x is equal to 2 raised to the power x is all positive real numbers that is open interval from 0 to infinity. So in general for the exponential function f of x is equal to a into b raised to the power x where b is greater than 0 or 0 is less than b is less than 1. Now if a is greater than 0 then the domain will be all real numbers that is open interval from minus infinity to infinity and range will be all positive real numbers or y is greater than 0 that is open interval from 0 to infinity. If the value of a is less than 0 then the domain will be all real numbers that is the open interval from minus infinity to infinity and range will be all negative real numbers or y is less than 0 that is open interval from minus infinity to 0. For example if we take the function f of x is equal to minus of 2 raised to the power x here we can see that the value of a is equal to minus 1 which is less than 0. Now this is the graph of the function f of x is equal to minus 2 raised to the power x. Now we can see here again that x can take any real value so its domain will be set of all real numbers that is from minus infinity to infinity. Now we can see that this curve lies below x axis that is towards negative y values so its range will be all negative real values that is y is less than 0 or open interval from minus infinity to 0. Now we will discuss exponential function of the form f of x is equal to a into v raised to the power x plus k. Now let us consider the graph of y is equal to 2 raised to the power x plus 3. Now if we look at this graph we can see that x can take any real value so its domain is all real numbers that is open interval from minus infinity to infinity. Now for the range we have to see y values. Here we see that the curve does not lie below y is equal to 3 also if we draw a horizontal line at y is equal to 3 the curve does not intersect this line. Here the curve is moving upwards for positive values of y greater than 3 so values of y can be from 3 to infinity so its range is all values greater than 3 or y is greater than 3 that is open interval from 3 to infinity so in general for exponential function of the form f of x is equal to a into b raised to the power x plus k. If the value of a is greater than 0 then domain is set as all real numbers that is open interval from minus infinity to infinity and range is all values greater than k or y is greater than k that is the open interval from k to infinity and if the value of a is less than 0 then domain is set of all real numbers that is from minus infinity to infinity and range is all values less than k or y is less than k that is open interval from minus infinity to k. Let us consider an example if f of x is equal to minus 20 into 4 raised to the power x minus 30 then we have to find domain and range of this function. Now here in this function we can see that the value of a is minus 20 which is less than 0 and the value of k is equal to minus 30 so here domain will be given by set of all real numbers that is from minus infinity to infinity now since the value of a is less than 0 so range will be all values less than k that is less than minus 30 or we can write it as y is less than minus 30 that is the open interval from minus infinity to minus 30 hence we have got the domain and range of this function. Here we should note that if the exponent is negative for example in y is equal to 2 raised to the power minus x then also domain and range remains same because the graph is reflected in y axis only also y is equal to 2 raised to the power minus x can be written as 1 by 2 whole raised to the power x so base b now lies between 0 and 1 so domain and range does not change thus in this session we have discussed domain and range of exponential functions this completes our session hope you enjoyed this session