AEF.SR: Subtraction and Roots
The relation of differences to roots is given by the simple but
important theorem that is a result of other basic field properties of
Theorem SR: For any $a,b \in R$, $a - b = 0$ if and only if $a=b$.
A simple corollary of Theorem SR: $f(x)=f_1(x) - f_2(x) =0$ if and only if $f_1(x)=f_2$.
For an elementary function, $f$ that can be expressed in a form with the
final step in its definition being the difference of two elementary
functions, $f_1$ and $f_2$, the corollary entails further that set of
the roots of $f$ is the set of roots of the equation, $f_1(x)=g(x)$.
A mapping diagram can visualize these results by visually connecting
coincident arrows in the mapping diagrams for $f_1$ and $f_2$ to a root
Example AEF.SR.1 Visualize the roots of $f(x)=(x^2-4) - \sin(x) $ for $x \in [-4,4]$.