We return to the notion of a function and examine the allowable inputs.

Introduction

We often think about functions as a process which transforms an input into some output. Sometimes that process is known to us (such as when we have a formula for the function) and sometimes that process is unknown to us (such as when we only have a small table of values).

a.
Suppose the quadratic function is given by . Are there any values that can’t be plugged into ?
b.
Suppose a square has side length denotes by the variable , and area denoted by . The area of the square is a function of the side length, . Are there any values of that don’t make sense?
c.
Suppose that is the rational function given by and that is the constant function given by . Are these the same function? Why or why not?

The Domain of a Function

When we are given a function, sometimes the domain is given to us explicitly. Consider the function for . The phrase “for ” tells us the domain for this function. We may be able to plug any number into the expression , but it’s only when that this gives our function. For instance, , but is undefined.

Sometimes, when we are given a function as a formula, we are not told the domain. In these circumstances we use the implied domain.

Interval Notation

As in the previous example, solutions of inequalities play an important role in expressing the domains of many types of functions. As a standard way of writing these solutions, we rely on interval notation. Interval notation is a short-hand way of representing the intervals as they appear when sketched on a number line. The previous example involved which, when sketched on a number line, is given by

This sketch consists of a single interval with left-hand endpoint at and no right-hand endpoint (it keeps going). In interval notation, this would be written as . This is an example of a closed infinite interval, “closed” because the point at (the only endpoint) is included and “infinite” because it has infinite width. The solid dot at indicates that the point is included in the interval.

There are five different types of infinite intervals: the first two are closed infinite intervals (which contain their respective endpoint) and the other three are open infinite intervals (which do not contain the endpoint). For a fixed real number , these are:

(a)
represents ,
(b)
represents ,
(c)
represents ,
(d)
represents , and
(e)
represents all real numbers.

The notation uses the square bracket to indicate that the endpoint is included and the round parenthesis to indicate that the endpoint is not included.

Not every interval is infinite, however. Consider the interval in the following sketch

which consists of all with . It is not an infinite interval, having endpoints at and . The endpoint at is not included, but the endpoint at is included. In interval notation this would be written as . As with the infinite intervals, the square bracket indicates that the right-hand endpoint is included and the round parenthesis endicates that the left-hand endpoint is not included. (This is an example of a “half-open interval”.)

For a bounded intervals (ones that are not infinite), there are also four possibilities. For and both fixed real numbers, these are:

(a)
represents ,
(b)
represents ,
(c)
represents and
(d)
represents .

Practically, this amounts to writing the left-hand endpoint, the right-hand endpoint, then indicating which endpoints are included in the interval.

The Domains of Famous Functions

Earlier you were introduced to the graphs of several “Famous Functions”. We will revisit these functions over and over again throughout our studies. For now, we will formalize what we have seen with their graphs.

Spotting Values not in the Domain

Of our list of famous functions, notice that only rational functions, radicals, and logarithms have domain that is not the full set of all real numbers, . When trying to find the domain of a function constructed out of famous functions, this gives us some guidelines to follow. The following list is not exhaustive, but gives a good place to begin.

Remember that the number zero is neither positive nor negative. The non-negative numbers are , while the positive numbers are .

Piecewise Defined Functions and Restricted Domains

Consider the function for , and the function (given without this restriction). The implied domain of is , but what can we say about ? The formula makes sense when , but the function definition for has the added statement “for ”. This is telling us the domain of is . In this case is undefined.

We can think of the function as coming from the function by deciding that some inputs are not valid. We have restricted the domain.

Suppose we have a function given by for (which has domain ) and a different function given by for (which has domain ). If we are given an -value in the interval , that input can only be plugged into one of these two functions. Let’s create a new function by setting if and by setting if . As a compact way of writing this, we would say:

The function above is a piecewise defined function. On the interval it is given by the formula , and on the interval it is given by the formula . It has two pieces, one piece is quadratic and the other piece is linear. The graph of the function is given below.