Example OAF.SAE.1 : Suppose $\frac {x^2 - 4}{x^2+1} = 0$. Find $x$.
Solution: Examining the numerator of the fraction of the equation, we see that it can be solved by solving $x^2-4 = 0 $. This gives the solutions that $x =  \pm 2$.

Comment: We can consider the expression on the left hand side of the equation as a rational function of $x$ giving $  f(x) = \frac {x^2 - 4}{x^2+1} $ . Now the problem can be restated: to find a $x$ where $f(x) = 0$. This problem and its solution can be visualized both on the graph and the mapping diagram for the function $f$. 
For the graph of $f$: Find $y=0$ on the Y axis , then find the point(s) on the graph of $f$ with second coordinate $0$, determine it's first coordinates, $2$ and $-2$, and those are the desired values for $x$. For the mapping diagram of $f$: Find $y=0$ on the target axis , then find point(s) $x$ on the source axis with the function arrow pointing to $0$.

To do this, look for the point(s) on the $X$ axis, the line $y=0$, where the axis intersects the graph of $f$ To do this, find the local extreme point of the quadratic numerator of $f$ on the mapping diagram, $x = 0$. Move symmetrically above and below that value by $\pm 2$, which are the desired values for $x$.
Martin Flashman, 5 December 2013, Created with GeoGebra