Introduction

In arithmetic, we execute processes where we take two numbers to generate a new number. For example, . The number results from adding the numbers and . Similarly, we can multiply two numbers to generate a new one: .

We can work similarly with functions. Just as we can add, subtract, multiply, and divide numbers, we can also add, subtract, multiply, and divide functions to create a new function from two or more given functions.

Algebra of Functions

In most mathematics up until calculus, the main object we study is numbers. We ask questions such as

  • “What number(s) form solutions to the equation ?”
  • “What number is the slope of the line represented by ?”
  • “What number is generated as output by the function by the input ?”

Certainly we also study overall patterns as seen in functions and equations, but this usually occurs through an examination of numbers themselves, and we think of numbers as the main objects being acted upon.

This changes in calculus. In calculus, the fundamental objects being studied are functions themselves. A function is a much more sophisticated mathematical object than a number, in part because a function can be thought of in terms of its graph, which is an infinite collection of ordered pairs of the form .

It is often helpful to look at a function’s formula and observe algebraic structure. For instance, given the quadratic function we might benefit from thinking of this as the sum of three simpler functions: the constant function , the linear function that passes through the point with slope , and the concave down basic quadratic function . Indeed, each of the simpler functions , , and contribute to making be the function that it is. Likewise, if we were interested in the function , it might be natural to think about the two simpler functions and that are being multiplied to produce .

We thus naturally arrive at the ideas of adding, subtracting, multiplying, or dividing two or more functions, and hence introduce the following definitions and notation.

We are thinking here about and being functions with real numbers as outputs. Performing these operations on the functions means applying the corresponding operation to the output values of the functions.

Consider the functions and . These are functions whose graphs are straight lines with slopes and respectively.

Let’s look at a few points on these graphs. Since and , the sum of those output values is , so we’ll mark the point . Similarly and , with so we’ll mark the point . As and , we’ll also mark .

Notice that the points we’ve marked , , and are starting to form a straight line. Let’s connect those dots to examine the line constructed this way. The graph obtained this way is the graph of . This graph is a straight line passing through the points and , so the line has equation .

For a given value of , we know . This means by combining the like terms. Notice that this aligns with the graph we found above. This example shows that we can work with these operations through formulas for our functions as well.

When we work in applied settings with functions that model phenomena in the world around us, it is often useful to think carefully about the units of various quantities. Analyzing units can help us both understand the algebraic structure of functions and the variables involved, as well as assist us in assigning meaning to quantities we compute. We have already seen this with the notion of average rate of change: if a function measures the population in a city in year and we compute , then the units on are “people per year,” and the value of is telling us the average rate at which the population changes in people per year on the time interval from year to year .

Let be a function that measures a car’s fuel economy in the following way. Given an input velocity in miles per hour, is the number of gallons of fuel that the car consumes per mile (i.e., “gallons per mile”). We know that .
(a)
What is the meaning of the statement “” in the context of the problem? That is, what does this say about the car’s fuel economy? Write a complete sentence.
(b)
Consider the function . What is the value of ? What are the units on ? What does measure?
(c)
Consider the function . What is the value of ? What are the units on ? What does measure?
(d)
Do , , and tell us fundamentally different information, or are they all essentially saying the same thing? Explain.
(e)
Suppose we also know that . Find the average rate of change of on the interval . What are the units on the average rate of change of ? What does this quantity measure? Write a complete sentence to explain.

Taking a complicated function and determining how it is constructed out of simpler ones is an important skill to develop. At the beginning of this section we split the function into the sum/difference of three simple functions, , , and . Let us experiment with splitting a few more complicated functions.

(a)
Find functions and so that can be written as .
(b)
Find functions and so that can be written as . Find two other choices for and .
(c)
If and are functions, we know that is the function given by . What function do you think the notation means? Find functions and so that can be written as .