Exponential Functions

Introduction and Summary

This chapter deals with radicals and exponential functions--functions that contain variable exponents. Here, the reader will review the meanings of negative and fractional exponents, learn how to solve equations containing radicals, and learn how to evaluate and graph exponential functions.

The first section reviews negative and fractional exponents. It explains how to evaluate expressions containing negative and fractional exponents. This material is also covered in Negative Exponents and Fractional Exponents.

The next section deals with equations containing radicals. Solving radical equations is similar to solving ordinary equations using inverse operations, but with two key differences--taking the inverse of a square leads to multiple solutions, and taking the inverse of a square root can lead to false solutions. This section explains how to find both solutions to an equation containing a square, and how to recognize and eliminate the false solutions to an equation containing a square root.

The final section introduces exponential functions. It explains how to graph an exponential function and how to find the domain and range of an exponential function. It also addresses translations, stretches, shrinks, reflections, and rotations of exponential functions.

Exponential functions are one of the many types of functions that mathematicians study. They are useful because they describe many real-world situations, including those in economics and in physics. In addition, they are interesting from a mathematical perspective because they employ the variable in an unusual way. While rational and polynomial functions multiply the variable by itself a fixed number of times, exponential functions vary the number of times a constant is multiplied by itself.