The Domain and Range

Published: Jul 21st, 2020

Learning Objectives
By the end of this lesson, you should be able to:
1. Write and interpret set notation

In the next section, we will be discussing a few need-to-know functions. To get the most out of that lesson, we must first understand the domain and range. You might recall in our first lesson, "Introduction to Relations", that the domain is the set of all x values for which a function is defined, and the range is the set of y values within the domain of a function for which \(y=f(x)\) is a real number.

We express the domain and range of functions using set notation. You have been introduced to set notation in the past, but, just to make sure, we will look at the anatomy of a set and see what each component means. This is summarized in the following diagram:

Now, while that's all fine and dandy for some functions, not all functions are defined everywhere, either for the domain, range, or both. For example, consider \(f(x) = x^2\), shown below:

This function, while defined for all x in the domain, is only defined for \(y>0\). We indicate the domain and range as \(D: \left\{x\in\mathbb{R}\right\}\)and \(R:\left\{y\in\mathbb{R}|y\geq0\right\}\), respectively. In set notation, we indicate restrictions using a pipe separator "|", followed by the restriction. This could be the range for which the set applies, or points that the set must not equal. 

To get you better acquainted with set notation, here are a few functions with restrictions, along with their respective domains and ranges. While you might not have seen these functions before, observe their graphs and see how this translates to set notation, then try a few practice problems for yourself. Note that in some cases, the function approaches, but never quite reaches some value. In such a case, the function can never equal that value, and it represents a restriction to either the domain or range. In other cases, we must know the properties of some mathematical operators to predict limits. For example: 

  • the square root of a negative number \(\sqrt{x},\:x<0\) is not a real number, so it represents a restriction to the domain or range;
  • the denominator of an inverse function \(\frac{1}{x}\) can never equal zero, and if the numerator is one (as in this case), the range can never equal 0, either.
  • as you will learn later, an exponential function is always greater than or equal to 0, because there is no number to which we can raise a number to get zero or a negative number. In the section on inverse functions, we will see how this translates to their inverse: the logarithmic functions.
  • sine and cosine functions, as you will learn later, do not have restrictions on the domain, but can only occur within \(-1\leq y\leq1\)

Note that these are only rules of thumb. In cases where we have transformations to the function, we must also transform their domains and ranges, as the functions themselves (and thus their restrictions) have changed. Don't worry if you feel a little bit out of your depth with some of these problems; by the time you're done this course, you will find them VERY simple!



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