Although disguised, inverse functions play an important role in the ordinary solution of equations. For example, consider the steps used to solve this simple algebra problem:

To go from the first equation to the second equation, let and , and let $f(x) = x – 4$. This is an bijective function with inverse . Therefore,

Stated another way,

Again, let $X_3 = 2x$ and $X_4 = 6$, and let $g(x) = x/2$. This is also a bijective function with inverse function $g^{-1}(x) = 2x$. Therefore,

So we are guaranteed that is the one and only one solution of this equation.

If the process of solving an equation requires the use of a function that isn’t a bijection, then funny things can happen. For example, consider the slightly more complicated equation

Let’s starting solving by squaring both sides:

So there are two solutions, right? Well…

,

but !

So what happened? In other words, what is qualitatively different about this problem that didn’t happen in the first problem to produce an extraneous solution? The problem is the first step. Let and . We applied the function to both sides. Unfortuntely, is not an invertible function when using the entire real line as the domain of . In other words,

implies ,

but does not imply that .

The practical upshot is that, when arriving at the final step of the solution, we can’t be certain that the “solutions” we obtain actually work. Instead, what we’ve really shown that anything other than the solutions can’t work, which is different than saying that these two solutions actually do work. So it remains to actually check that these potential solutions are actually solutions (or not).

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.
View all posts by John Quintanilla

Published

One thought on “Inverse Functions: Solving Equations (Part 4)”

## One thought on “Inverse Functions: Solving Equations (Part 4)”