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.
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