Active Calculus 2nd Ed - HU version

In this section, we provide a formal definition of a function and examine several ways in which functions are represented—namely, through tables, formulas, and graphs. We study formal notation and terms related to functions. We also define composition of functions and symmetry properties. Most of this material will be a review for you, but it serves as a handy reference to remind you of some of the algebraic techniques useful for working with functions.

Given two sets \(A\) and \(B\text{,}\) a set with elements that are ordered pairs \((𝑥,𝑦)\text{,}\) where \(𝑥\) is an element of \(A\) and \(𝑦\) is an element of \(B\text{,}\) is a relation from \(A\) to \(B\text{.}\) A relation from \(A\) to \(B\) defines a relationship between those two sets. A function is a special type of relation in which each element of the first set is related to exactly one element of the second set. The element of the first set is called the input; the element of the second set is called the output. Functions are used all the time in mathematics to describe relationships between two sets. For any function, when we know the input, the output is determined, so we say that the output is a function of the input. For example, the area of a square is determined by its side length, so we say that the area (the output) is a function of its side length (the input). The velocity of a ball thrown in the air can be described as a function of the amount of time the ball is in the air. The cost of mailing a package is a function of the weight of the package. Since functions have so many uses, it is important to have precise definitions and terminology to study them.

For example, consider the function \(f\text{,}\) where the domain is the set of all real numbers and the rule is to square the input. Then, the input \(x=3\) is assigned to the output 32=9. Since every nonnegative real number has a real-value square root, every nonnegative number is an element of the range of this function. Since there is no real number with a square that is negative, the negative real numbers are not elements of the range. We conclude that the range is the set of nonnegative real numbers.

For a general function \(f\) with domain \(D\text{,}\) we often use \(x\) to denote the input and \(y\) to denote the output associated with \(x\text{.}\) When doing so, we refer to \(x\) as the independent variable and \(y\) as the dependent variable, because it depends on \(x\text{.}\) Using function notation, we write \(y=f(x)\text{,}\) and we read this equation as “\(y\) equals \(f\) of \(x\text{.}\)” For the squaring function described earlier, we write \(f(x)=x^2\text{.}\)

We can also visualize a function by plotting points \((𝑥,𝑦)\) in the coordinate plane where \(𝑦=𝑓⁡(𝑥)\text{.}\) The graph of a function is the set of all these points. For example, consider the function \(𝑓\text{,}\) where the domain is the set \(𝐷=\{1,2,3\}\) and the rule is \(𝑓⁡(𝑥)=3−𝑥\text{.}\) In Figure 1.1.4, we plot a graph of this function.

Every function has a domain. However, sometimes a function is described by an equation, as in \(𝑓⁡(𝑥)=𝑥^2\text{,}\) with no specific domain given. In this case, the domain is taken to be the set of all real numbers \(𝑥\) for which \(𝑓⁡(𝑥)\) is a real number. For example, since any real number can be squared, if no other domain is specified, we consider the domain of \(𝑓⁡(𝑥)=𝑥^2\) to be the set of all real numbers. On the other hand, the square root function \(𝑓⁡(𝑥)=\sqrt{𝑥}\) only gives a real output if \(𝑥\) is nonnegative. Therefore, the domain of the function \(𝑓⁡(𝑥)=\sqrt{𝑥}\) is the set of nonnegative real numbers, sometimes called the natural domain.

Tables

Functions described using a table of values arise frequently in real-world applications. Consider the following simple example. We can describe temperature on a given day as a function of time of day. Suppose we record the temperature every hour for a 24-hour period starting at midnight. We let our input variable 𝑥 be the time after midnight, measured in hours, and the output variable 𝑦 be the temperature 𝑥 hours after midnight, measured in degrees Fahrenheit. We record our data in Figure 1.1.7.

We can see from the table that temperature is a function of time, and the temperature decreases, then increases, and then decreases again. However, we cannot get a clear picture of the behavior of the function without graphing it.

Graphs

Given a function 𝑓 described by a table, we can provide a visual picture of the function in the form of a graph. Graphing the temperatures listed in Table 1.1 can give us a better idea of their fluctuation throughout the day. Figure 1.6 shows the plot of the temperature function.

From the points plotted on the graph in Figure 1.6, we can visualize the general shape of the graph. It is often useful to connect the dots in the graph, which represent the data from the table. In this example, although we cannot make any definitive conclusion regarding what the temperature was at any time for which the temperature was not recorded, given the number of data points collected and the pattern in these points, it is reasonable to suspect that the temperatures at other times followed a similar pattern, as we can see in Figure 1.7.

Algebraic Formulas

Sometimes we are not given the values of a function in table form, rather we are given the values in an explicit formula. Formulas arise in many applications. For example, the area of a circle of radius 𝑟 is given by the formula 𝐴⁡(𝑟)=𝜋⁢𝑟2. When an object is thrown upward from the ground with an initial velocity 𝑣0 ft/s, its height above the ground from the time it is thrown until it hits the ground is given by the formula 𝑠⁡(𝑡)=−16⁢𝑡2+𝑣0⁢𝑡. When 𝑃 dollars are invested in an account at an annual interest rate 𝑟 compounded continuously, the amount of money after 𝑡 years is given by the formula 𝐴⁡(𝑡)=𝑃⁢𝑒𝑟⁢𝑡. Algebraic formulas are important tools to calculate function values. Often we also represent these functions visually in graph form.

Given an algebraic formula for a function 𝑓, the graph of 𝑓 is the set of points (𝑥,𝑓⁡(𝑥)), where 𝑥 is in the domain of 𝑓 and 𝑓⁡(𝑥) is in the range. To graph a function given by a formula, it is helpful to begin by using the formula to create a table of inputs and outputs. If the domain of 𝑓 consists of an infinite number of values, we cannot list all of them, but because listing some of the inputs and outputs can be very useful, it is often a good way to begin.

When creating a table of inputs and outputs, we typically check to determine whether zero is an output. Those values of 𝑥 where 𝑓⁡(𝑥)=0 are called the zeros of a function. For example, the zeros of 𝑓⁡(𝑥)=𝑥2−4 are 𝑥=±2. The zeros determine where the graph of 𝑓 intersects the 𝑥-axis, which gives us more information about the shape of the graph of the function. The graph of a function may never intersect the x-axis, or it may intersect multiple (or even infinitely many) times.

Another point of interest is the 𝑦-intercept, if it exists. The 𝑦-intercept is given by (0,𝑓⁡(0)).

Since a function has exactly one output for each input, the graph of a function can have, at most, one 𝑦-intercept. If 𝑥=0 is in the domain of a function 𝑓, then 𝑓 has exactly one 𝑦-intercept. If 𝑥=0 is not in the domain of 𝑓, then 𝑓 has no 𝑦-intercept. Similarly, for any real number 𝑐, if 𝑐 is in the domain of 𝑓, there is exactly one output 𝑓⁡(𝑐), and the line 𝑥=𝑐 intersects the graph of 𝑓 exactly once. On the other hand, if 𝑐 is not in the domain of 𝑓,𝑓⁡(𝑐) is not defined and the line 𝑥=𝑐 does not intersect the graph of 𝑓. This property is summarized in the vertical line test.

We can use this test to determine whether a set of plotted points represents the graph of a function (Figure 1.1.11).

Now that we have reviewed the basic characteristics of functions, we can see what happens to these properties when we combine functions in different ways, using basic mathematical operations to create new functions. For example, if the cost for a company to manufacture 𝑥 items is described by the function 𝐶⁡(𝑥) and the revenue created by the sale of 𝑥 items is described by the function 𝑅⁡(𝑥), then the profit on the manufacture and sale of 𝑥 items is defined as 𝑃⁡(𝑥)=𝑅⁡(𝑥)−𝐶⁡(𝑥). Using the difference between two functions, we created a new function.

When we compose functions, we take a function of a function. For example, suppose the temperature 𝑇 on a given day is described as a function of time 𝑡 (measured in hours after midnight) as in Table 1.1. Suppose the cost 𝐶, to heat or cool a building for 1 hour, can be described as a function of the temperature 𝑇. Combining these two functions, we can describe the cost of heating or cooling a building as a function of time by evaluating 𝐶⁡(𝑇⁡(𝑡)). We have defined a new function, denoted 𝐶∘𝑇, which is defined such that (𝐶∘𝑇)⁡(𝑡)=𝐶⁡(𝑇⁡(𝑡)) for all 𝑡 in the domain of 𝑇. This new function is called a composite function. We note that since cost is a function of temperature and temperature is a function of time, it makes sense to define this new function (𝐶∘𝑇)⁢(𝑡). It does not make sense to consider (𝑇∘𝐶)⁢(𝑡), because temperature is not a function of cost.

A composite function 𝑔∘𝑓 can be viewed in two steps. First, the function 𝑓 maps each input 𝑥 in the domain of 𝑓 to its output 𝑓⁡(𝑥) in the range of 𝑓. Second, since the range of 𝑓 is a subset of the domain of 𝑔, the output 𝑓⁡(𝑥) is an element in the domain of 𝑔, and therefore it is mapped to an output 𝑔⁡(𝑓⁡(𝑥)) in the range of 𝑔. In Figure 1.12, we see a visual image of a composite function.