Exponential Functions
Exponential functions differ from power functions in that the exponent itself is the variable. For example:
| x - variable | n - constant | ||
| The range is the positive real numbers. |
xn - power function nx - exponential function
Exponential functions are encountered through out math. They are sometimes introduced in algebra because a student taking precalculus or calculus will need to know them.
The most common exponential function seen in math is ex. The base of the function is the natural number symbolized by e which is an irrational number approximated. e = 2.7182818284590...
Definition of an Exponential Function
The form of an exponential function is formally defined as:
| f(x) = bx | |
| The domain is the set of real numbers. | |
| The range is the positive real numbers. |
The exponential function dominates the power function because it increases more rapidly as the value of the exponent approaches infinity. In the beginning, however, the power function increases faster. This can be seen on a graph of the two functions.
Properties of Exponential Functions
Some of the properties of a exponential function are the following:
- The graph is always increasing with the exponent.
- When x goes to infinity then f(x) goes to infinity.
- The x-axis acts as a horizontal asymptote.
- The function f(x) = bx passes through the point (0,1).
The function ex is the natural exponential function. The slope of the line that is tangent to ex at (0,1) is one, m = 1. Working with exponential functions follow the same rules for exponents as do power functions. Some of the applications for exponential functions are in growth and decay models. A population of bacteria that doubles every hour can be modeled using them. Radioactive decay can also be modeled when the exponent is negative.
