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).

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.

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