An exponential function is a mathematical function where the variable appears in the exponent. The general form is f of x equals a to the power of x, where a is a positive constant not equal to one, and x is the variable. Here we see the graph of f of x equals two to the power of x, which shows the characteristic exponential growth curve.
Exponential functions have several important properties. Their domain includes all real numbers, but their range is always positive. Every exponential function passes through the point zero comma one. When the base is greater than one, the function is increasing, but when the base is between zero and one, the function is decreasing. All exponential functions have a horizontal asymptote at y equals zero.
There are several common exponential functions we encounter frequently. The function f of x equals two to the power of x represents doubling. The natural exponential function f of x equals e to the power of x uses Euler's number e, approximately two point seven one eight. The function f of x equals ten to the power of x represents powers of ten. We can also have scaled exponential functions like three times two to the power of x.
Exponential functions have many important real-world applications. Population growth follows the formula P of t equals P naught times e to the power of r t. Compound interest uses A equals P times one plus r to the power of t. Radioactive decay follows N of t equals N naught times e to the negative lambda t. These functions also model bacterial growth and investment returns, making them essential tools in science, finance, and biology.
To summarize what we have learned about exponential functions: they have the form f of x equals a to the power of x, where the base a must be positive and not equal to one. These functions demonstrate exponential growth or decay patterns and are widely used in modeling population growth, financial calculations, and scientific phenomena.