Operations with Functions
Introduction and Summary
In this chapter, the student will learn how to perform basic operations with functions--addition, subtraction, multiplication, and composition--as well as how to find the inverse of a function.
Whenever a new entity is introduced, such as matrices, polynomials, or, in this case, functions, we learn how to perform addition with that entity. Addition of functions is the focus of the first section. We will first learn how to evaluate functions, and then we will learn how to add them with specific inputs and in the general case. As a byproduct of addition, we will also learn subtraction of functions.
The next step, as usual, is to learn how to perform multiplication with the new entity. The second section explains how to multiply a function by a scalar and how to multiply a function by another function. This section also explains how to compute a compound function; that is, a function of another function.
Like real numbers and matrices, functions have inverses. The next section gives the definition of inverse functions, and explains how to find them by reversing the operations of the function.
The final section presents two more methods for finding the inverse of a function. The first method involves substituting x for f (x) , substituting f -1(x) for x , and isolating f -1(x) . The second method involves graphing the function and reflecting it over the line y = x . All three methods of finding inverses are useful, and they can be used in combination to check each other.
Since functions are such an important part of algebra and calculus, it is crucial to understand how to perform basic operations with them. This is the primary goal of this chapter.