AEF.MR: Multiplication [Division] and Roots
The relation of products to roots is given by the simple but
important theorem that is a result of other basic field properties of
real numbers.
Theorem MR: If $a,b \in R$ and $a\cdot b = 0$, then either $a=0$ or $b=0$.
A simple corollary of Theorem MR: If $f(x)=f_1(x)\cdot f_2(x) =0$ then either $f_1(x)=0$ or $f_2(x) =0$.
For an elementary function, $f$ that can be expressed in a form with the
final step in its definition being the product of two elementary
functions, $f_1$ and $f_2$, the corollary entails further that set of
the roots of $f$ is the union of the set of roots of the simpler
elementary functions, $f_1$ and $f_2$.
A mapping diagram can visualize these results by visually connecting
roots in mapping diagrams for $f_1$ and $f_2$ to a root of $f$.
Example AEF.MR.1 Visualize the roots of $f(x)=(x^2-4)\cdot\sin(x) $ for $x \in [-4,4]$.
The relation of quotients to roots is given by the simple but
important theorem that is a result of other basic field properties of
real numbers.
Theorem DR: For $a,b \in R$ , $b \ne 0$, $a\div b = 0$ if and only if $a=0$.
A simple corollary of Theorem DR: For $f(x)=f_1(x)\div f_2(x), f_2(x) \ne 0, f(x)=0$ if and only if $f_1(x)=0$.
For an elementary function, $f$ that can be expressed in a form with the
final step in its definition being the quotient 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 function, $f_1$.
Example AEF.DR.1 Visualize the roots of $f(x)=(x^2-4)\div \sin(x) $ for $x \in [-4,4]$.