Beginning and intermediate algebra spend a considerable amount of time solving
equations without any reference to functions.
As a result, very little time is spent of visualizing equations or how
they are solved.
In more advance courses before calculus, functions are
introduced, but still little is done to make sense visually of the
solution of equations beyond connecting to intersections of curves with
other curves or with the x-axis.
In almost every section of this resource, visualizing the solution of an
equation has been connected to one or more core functions and examples
using composition with linear functions. Though almost every equation
studied before calculus involves elementary functions, solving
exactly an equation formed with elementary functions is a difficult task
for which there are
no universal techniques.
For instance, the fundamental theorem of algebra, OAF.FTA, indicates
that every real non-constant polynomial equation will have at
least one (possibly complex) root. Students in algebra courses are quite
familiar with solving linear and quadratic equations using the
coefficients of a polynomial to express the solution(s).
However, for an arbitrary fifth
degree real polynomial equation there is no single formula involving
only radicals related to the coefficients of the polynomial that
produces a solution. [See the Abel-Ruffini theorem, Wikipedia]
When no exact solution for an equation is possible, there are some general numerical
techniques for estimating a solution which are still evolving as technology improves to make these
tools more effective and efficient. [See Root-finding algorithm, Wikipedia]
In this subsection we begin with links for review of the previous subsections
on solving equations.
We then highlight mapping diagrams that make connections to the
recursive way that elementary functions are defined, shedding more
light on solving these equations.
A discussion of visualizing numerical techniques can be found in Subsection AEF.NSEq.
Review: Subsections on Solving Equations
LF.LEq Solving Linear Equations with Linear Functions
QF.QEq Solving Quadratic Equations with Quadratic Functions
OAF.SAE Solving Equations That Use Other Algebraic Functions
ELF.SEQ Solving Exponential and Logarithmic Equations
TRIG.TrigEq Solving Trigonometric Equations
Visualizing Connections for Elementary Functions