I am currently in a Pre Calculus class at my High School. I have come across the concept of extraneous solutions, particularly when solving absolute value equations, radical equations, and logarithmic equations. My question is, why do these solutions exist?
My teacher never explained this, which is understandable given that I am in a High School math class, and there isn't much time for the teacher to go into the actual derivations of everything. I am wondering because I plan to major in Mathematics, and having a conceptual understanding of this is important to me.
If anyone could explain the reason extraneous solutions exist for the three examples I noted, it would be of great help to me.
Best Answer
One reason extraneous solutions exist is because some functions are not injective. In particular, given an equality
$$a(x)=b(x)$$
we can square both sides to get
$a(x)^2 = b(x)^2$. It is true that $a(x)=b(x) \Longrightarrow a(x)^2 = b(x)^2$; however, it is not true that $a(x)^2 = b(x)^2 \Longrightarrow a(x)=b(x)$. In particular, we may have $a(x) = -b(x)$.
Consequently, we may have solutions $x$ to the equation $a(x)^2 = b(x)^2$ which satisfy $a(x) = -b(x)$ but not $a(x) = b(x)$. These are extraneous. In general, you will not lose solutions squaring both sides, but gain extraneous ones.
Another reason is domain considerations. Consider the logarithmic equation
$\log(f(x)) + \log(g(x)) = \log(h(x))$
We thus solve $f(x) \cdot g(x) = h(x)$. We might get a bunch of solutions $x$ to this equation, but they are only valid so long as $f(x)$ and $g(x)$ are greater than zero, because the identity $\log(x) + \log(y) = \log(xy)$ only works for positive $x,y$. However, again, you will not lose solutions with this method. If $x$ satisfies $\log(f(x)) + \log(g(x)) = \log(h(x))$, then that $x$ also must satisfy $f(x) \cdot g(x) = h(x)$.
You have to remember what you do when you solve an equation. It is easy to forget since it is almost second nature to us. What you do is alter the given equation such that solutions to the equation are preserved. When adding and subtracting, there is never a problem, since adding and subtracting are bijective maps. Other functions may not be as nice, and your altered equation may yield solutions which do not solve your original equation.