note that f ( x ) = ( ) = whereas f ( ) = = . Clearly f ( x ) is much greater than f ( x ). It is not difficult to prove that for all x > , f ( x ) > f ( x ). But we will not attempt to give a proof of this here.
Similarly, by choosing large values of x , one can verify that f ( x ) grows faster than f n ( x ) for any positive integer n . Definition The exponential function with positive base b > is the function y = f ( x ) = b x The graph of y = x is given in the Fig . . It is advised that the reader plots this graph for particular values of b like , and .
Following are some of the salient features of the exponential functions: ( ) Domain of the exponential function is R , the set of all real numbers. ( ) Range of the exponential function is the set of all positive real numbers. ( ) The point ( , ) is always on the graph of the exponential function (this is a restatement of the fact that b = for any real b > ). ( ) Exponential function is ever increasing; i.e., as we move from left to right, the graph rises above.
Fig . ( ) For very large negative values of x , the exponential function is very close to . In other words, in the second quadrant, the graph approaches x -axis (but never meets it). Exponential function with base is called the common exponential function .
In the Appendix A. . of Class XI, it was observed that the sum of the series ... !
! is a number between and and is denoted by e . Using this e as the base we obtain an extremely important exponential function y = e x .