An exponential function is a function in which the independent variable is an exponent. Exponential functions have the general form y = f (x) = a ^{x} , where a > 0 , a≠1 , and x is any real number. The reason a > 0 is that if it is negative, the function is undefined for -1 < x < 1 . Restricting a to positive values allows the function to have a domain of all real numbers. In this example, a is called the base of the exponential function.
Here is a little review of exponents:
exponent
a ^{-x} = . |
a ^{x+y} = a ^{x}×a ^{y}. |
a ^{x-y} = . |
a ^{0} = 1. |
a ^{x} = a ^{y};if and only if;x = y. |
Below are pictured functions of the form y = f (x) = a ^{x} and y = f (x) = a ^{-x} . Study them.
The domain of exponential functions is all real numbers. The range is all real numbers greater than zero. The line y = 0 is a horizontal asymptote for all exponential functions. When a > 1 : as x increases, the exponential function increases, and as x decreases, the function decreases. On the other hand, when 0 < a < 1 : as x increases, the function decreases, and as x decreases, the function increases.
Exponential functions have special applications when the base is e . e is a number. Its decimal approximation is about 2.718281828 . It is the limit approached by f (x) when f (x) = (1 + )^{x} and x increases without bound. Go ahead and plug the equation into your calculator and check it out. e is sometimes called the natural base, and the function y = f (x) = e ^{x} is called the natural exponential function.
The natural exponential function is especially useful and relevant when it comes to modeling the behavior of systems whose relative growth rate is constant. These include populations, bank accounts, and other such situations. Let the growth (or decay) of something be modeled by the function f (x) , where x is a unit of time. Let its relative growth rate ( ) be the constant k . Then its growth is modeled by the exponential function f (x) = f (0)e ^{kx} . Given any two of the following values: f (0) , k , or x , the third can be calculated using this function. In Applications we'll see some useful applications of this function.