Exponential Function

The function ex (where e is the number approximately 2.718) is called exponential function. The graph of the function is as follows.



We can observe that the graph of y = ex is upward-sloping, and increases faster as x increases. The graph always lies above the x-axis but can get arbitrarily close to it for negative x; thus, the x - axis is a horizontal asymptote. Exponential functions can be generalized as functions of the form f(x) = bx for a fixed base b which could be any positive real number. Exponential functions are characterized by the fact that their rate of growth is proportional to their value. For example, suppose we start with a population of cells such that its growth rate at any time is proportional to its size. The number of cells after t years will then be a (an exponential function) for some a > 0.

The shape of the graph of y = bx depends on whether b < b =" 1,"> 1 as shown below. The red graph is the graph of bx (b > 1), the blue graph is the graph of 1x, and the green graph is the graph of (1/b) x (b < 1).