# Inverse functions: Square root (Part 1)

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 $\sqrt{x^2}$.

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

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 $\sqrt{x^2} = x$. This is clearly incorrect if $x$ is negative:

$\sqrt{(-3)^2} = \sqrt{9} = 3 \ne -3$

The second follow-up mistake is then often mistake made by attempting to rectify the first mistake by writing $\sqrt{x^2} = \pm x$. The student usually intends the symbol $\pm$ to mean “plus or minus, depending on the value of $x$,” 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 $\sqrt{9}$ with a calculator, the calculator doesn’t flash between $3$ and $-3$; it returns only one answer.

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

$\sqrt{x^2} = |x|$

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.