In the previous posts of this series, I carefully considered the definition of . Let’s now repeat this logic to consider the definition of , where is an integer. We begin with even.

A typical case is ; the graph of is shown below.

The full graph of fails the horizontal line test if is even. Therefore, we need to apply the same logic that we used earlier to define . In particular, we essentially erase the left half of the graph. By restricting the domain to , we create a new function that does satisfy the horizontal line test, so that the graph of is found by reflecting through the line .

Written in sentence form,

If is even, then means that and . In particular, this is impossible for real if .

We now turn to the case of odd. Unlike before, the full graph of (in thick blue) satisfies the horizontal line test. Therefore, there is no need to restrict the domain to define the inverse function. (shown in thin purple).

In other words,

If is odd, then means that . There is no need to give a caveat on the possible values of .

In particular, and are both undefined since there is no (real) number so that or . However, is defined and is equal to since .

In this post, I take a deeper look at the standard mistake of “simplifying” incorrectly as just .

In the previous post, we noted that the function defined by fails the horizontal line test and thus does not have an inverse function.

However, we can restrict the domain of to make a new function that does satisfy the horizontal line test. This new function is identical to where both and are defined. Following tradition, we restrict the domain to :

defined by .

So by essentially erasing the left half of the parabola, we form a function that passes the horizontal line test which therefore has an inverse. Naturally, this function is . When I teach Precalculus, I like to write this as a sentence:

means that and .

Since and are inverse functions, it’s always true that and whenever these functions are defined. For example,

and

In other words, we are guaranteed that is always equal to — as long lies in the domain of … or, in other words, as long as is nonnegative.

Because if is negative, all bets are off.

Remember, the original function does not have an inverse. In particular, and are not inverse functions, and so it’s possible for to be something other than . For example,

, but

Of course, I don’t expect my Precalculus students to remember the subtle reason that this fails when they do their homework problems. But I do tell my Precalculus students that the nontrivial simplification of is a natural consequence of restricting the domain of a function that does not pass the horizontal line test to define an inverse function. In this example, as long as . However, if , then the result really could be just about anything else. For the current example, we have the pairwise simplification

if ;

if .

The last line is often uncomfortable for students, and so I remind them that is assumed to be negative so that is positive. Of course, there’s a new notation that mathematicians have developed so that this two-line simplification of can be compressed to a single line:

Unfortunately, in my opinion, this remains the problem that can be stated in two seconds or less (“Simplify the square root of squared”) that, in my opinion, is missed most often by mathematics students.

With the last four posts as prelude, let’s consider finding the inverse of the function defined by .

Of course, we can’t find an inverse for this function as stated. Colloquially, the graph of fails the horizontal line test. More precisely, there exist two numbers and so that but . (Indeed, there are infinitely many such pairs — and for any — but that’s beside the point.)

So how will we find the inverse of ? Well, we can’t. But we can do something almost as good: we can define a new function that’s going look an awful lot like . Here’s the function:

defined by .

Notice that the function looks an awful lot like , except that the domain has been restricted. In this way, the left half of the graph of is essentially erased to produce the graph of .

So while the graph of fails the horizontal line test, the graph of does pass the horizontal line test. Therefore, has an inverse function. We reverse the roles of domain and range (of course, for this example, the domain and range are identical):

Also, the graph of can be produced by reflecting through the line , producing the purple graph below.

Of course, the function is more customarily written as . Stated another way using the restricted domain of .

means that and .

Pedagogically, when teaching Precalculus, I’ve found that this way of writing the definition of is a useful prelude to the definition of the inverse trigonometric functions.

A couple of notes:

1. Notice that, if the purple parabola was completed, the full parabola would violate the vertical line test and thus fail to be a function. Thinking back to the original function, that’s another way of saying that the original full parabola violates the horizontal line test.
2. Since and are inverse functions, it’s always true that and . However, and are not inverse functions, and so it’s possible for to be something other than . This subtle distinction will be discussed more in tomorrow’s post.
3. Restricting the domain to was a perfectly arbitrary decision. It would have been perfectly OK if we have chosen the left half of the original parabola instead of the right half, as either half of the parabola satisfies the horizontal line test. So why did we choose the right half (with the nonnegative domain) instead of the left half (with the nonpositive domain)? That I can answer with one word: tradition. (By the way, finding an expression for the restriction of to is a standard problem in a first course in real analysis.

Here is one of the questions that I ask my class of future secondary mathematics teachers to answer.

A student asks, “My father was helping me with my homework last night and he said the book is wrong. He said that and , because and . But the book says . He wants to know why we are using a book that has mistakes.”

This is a very similar question to the simplification of , which was discussed in yesterday’s post. My experience has been that the above misconceptions involves confusion surrounding two very similar-sounding questions.

Question #1: Find all values of so that .

Question #2: Find the nonnegative value of so that .

The first question can be restated as solving , or finding all roots of a second-order polynomial. Accordingly, there are two answers. (Of course, the answers are and , written more succinctly as .) The second question asks for the positive answer to Question #1. This positive answer is defined to be , or .

In other words, it’s important to be sure that you’re answering the right question.

Is it all that important that is chosen to be the nonnegative solution to Question #1? Not really. We could have easily chosen the negative answer. The reason we choose the positive answer and not the negative answer can be answered in one word: tradition.

We want to be a well-defined function that produces only one output value, and there’s no mathematically advantageous reason for choosing the nonnegative answer aside from the important consideration that everyone else does it that way. And though students probably won’t remember this tidbit of wisdom when the time comes, the same logic applies when choosing the range of the inverse trigonometric functions.

Of course, for the present case, it totally makes sense to take the positive, less complicated answer as the output of . However, this won’t be as readily apparent when we consider the inverse trigonometric functions.

What is the math question that can be stated in two seconds but is most often answered incorrectly by math majors? In my opinion, here it is:

Simplify .

In my normal conversational voice, I can say “Simplify the square root of squared” in a shade less than two seconds.

Here’s a thought bubble if you’d like to think about this before I give the answer.

A common mistake made by algebra students (and also math majors in college who haven’t thought about this nuance for a while) is thinking that . This is clearly incorrect if is negative:

The second follow-up mistake is then often mistake made by attempting to rectify the first mistake by writing . The student usually intends the symbol to mean “plus or minus, depending on the value of ,” whereas the true meaning is “plus or minus” without any caveats. I usually correct this second mistake by pointing out that when a student finds with a calculator, the calculator doesn’t flash between and ; it returns only one answer.

After clearing that conceptual hurdle, students can usually guess the correct simplification:

In this series of posts, I’d like to expand on the thoughts above to consider some of the inverse functions that commonly appear in secondary mathematics: the square-root function and the inverse trigonometric functions.