Solving Equations That Use Other Algebraic Functions
Rational Functions in Equations
An equation involving only rational functions can always be
expressed in the form: $R(x) = \frac {P(x)}{Q(x)}= 0$.
A solution, $x=a$, must also be a solution for the polynomial
equation $P(x) = 0$.
Thus solving rational function equations reduces to solving
polynomial equations.
The Factor Theorem gives
the result that connects solving polynomial equation with the
factors of a polynomial functions.
Beginning algebra courses spend a considerable amount of time solving quadratic
equations without any reference to functions.
Solving equations with linear fractional expressions is also a
subject of beginning and intermediate algebra courses.
Other Algebraic Functions in Equations
Equations involving roots can be more challenging the rational
functions. They are more likely to be considered in courses at least
at the intermediate algebra level.
Here are some examples that illustrate the visualization of equation
solutions with mapping diagrams and graphs.
You can use this next dynamic example to visualize solving algebraic
equations like those in Examples QEQ.1 and QEQ.2 with a mapping
diagram and graph.
Example OAF.DSAE.0
Dynamic Views for solving an equation
$f(x) = g(x)$ and $f(x) = \frac {P(x)}{Q(x)}$ on Graphs and
Mapping Diagrams