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,
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,
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
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.