Exploring Functions

Contents

In this chapter, we give a detailed discussion of functions of a single independent variable. Functions are a hugely important concept in mathematics and will be used extensively in all that follows in this book. We begin with a general discussion of functions in the broad sense, and then proceed to discuss particular fundamental classes of functions: polynomial, rational, exponential, logarithmic, and circular (trigonometric) functions. These fundamental classes of functions will form the “building blocks” for the expressions that arise in the actuarial and financial context.

A crucial aim of this chapter is to encourage you to think of functions as mathematical objects and to be able to explore their properties. You should assume that we are working with the real numbers in all that follows.

2.1 General Properties and Methods

2.1.1 Mappings

Chapter 1 ended with a discussion of the expression

x+4)2

After studying the previous chapter, it should be clear that Eq. (2.1) is not an example of an equation or an identity; that is, it does not have an equal sign separating a LHS and RHS. Nor is it an inequality. In fact, the expression is an example of a function. As we shall see, functions have a precise meaning and not all expressions are necessarily functions. We now build toward an understanding of what is meant by the term “function.”

For the moment, you should think of Eq. (2.1) as a “rule” that returns an output for a given value of x as an input. For example, particular input and output values are stated in Table 2.1.

With the results of Table 2.1 in mind, it makes sense to think of Eq. (2.1) as a mapping. For example, in this particular case, − 4 has been mapped to 0, 1 has been mapped to 25, and 4 has been mapped to 64. Alternatively, using the “goes to” notation listed in Table 1.1, we can write these particular mappings as

4→(x+4)20,1→(x+4)225,4→(x+4)264

Rather than always writing the full expression that defines the mapping, it is convenient to label it with a single letter. For example, we could denote Eq. (2.1) by the letter f and write

:x→(x+4)2

In fact, the mapping is fully defined by the statement

:x→(x+4)2,∀x∈R

which is read as

It is often convenient to write this simply as

(x)=(x+4)2

which states the mapping’s label, f, and the symbol used to define the independent variable, x. Of course, the shorthand notation f(x) gives no information about the properties of the independent variable, for example, that this f is defined for all real numbers. However, properties of the input will typically be known from the context of the problem.

There is nothing special about using the letter f to denote a mapping and x to denote the independent variable. For example, the following statement defines exactly the same mapping as in Eq. (2.2)

:y→(y+4)2,∀y∈R

and can be summarized as g(y) = (y+4)2.

The set of possible values of the independent variable on which the mapping is defined is called the domain. For example, the domain of mapping (2.2) is the set of real numbers. However, the mapping

:z→(z+2)3,∀z∈[−2,2]

has a domain formed from the closed interval of real numbers between − 2 and 2 (including the end points).

If we understand that the domain defines the extent of the input of a mapping, the range defines the extent of the output of the domain under that mapping. For example, the range of mapping (2.2) is +∪{0}, equivalently 0,∞), and the range of function (2.3) is h(z) ∈ [0,64]. We can interpret this as it being impossible to find an input within the domain that gives an output outside of the range. We return to question of how to determine the mapping’s range later...