Mapping Diagrams Are Important.
The mapping diagram visualization may not be as familiar to you
as other methods for understanding functions. It is currently given
very little time in introductory presentations of the function
concept and even less as functions are explored in mathematics
courses through calculus and differential equations.
However, the use of mapping diagrams can provide a superb tool for
understanding functions at every level.
From the earliest experiences with linear function to college level
courses in linear algebra and beyond, the visualization of basic
concepts such as rate, composition and inverse functions are made
simple with mapping diagrams.
The core algebra functions of powers, roots, exponential and
logarithms gain much from the mapping diagram approach, as do the
trigonometric functions and their inverses when connected to the
unit circle characterization.
The basic premise is clear when one understands how easily these
diagrams visualize compositions
and inverses.
These two concepts are dominant in using functions everywhere.
For example -a quadratic polynomial function $q(x) = A x^2 +Bx + C$
can always be represented and visualized as the composition of
linear functions with the core function of $x^2$, so $q(x) = A
(x-h)^2 +k$.
This approach of using compositions is also prominent in studying
- rational functions, such as $ \frac A {x-h} + k$;
- the exponential and logarithmic functions,such as $ A *e^{Bx}$
and $A*\ln(Bx +C)$;
- trigonometric functions such as $A* trig( Bx+C)$;
- inverse trigonometric functions, such as $arcsin = \sin^{-1}$.
In other words: Using mapping diagrams can enrich the visualization
and understanding of all the elementary functions.
Solving equations and estimations also find added value from a
mapping diagram visualization.
The list of topics where mapping diagrams add insight and
understanding goes on to understanding both the derivative and the
definite integral.
You will see more as you proceed to investigate the sections of this
resource that match you own interests.
So, starting with beginning algebra and going as far as
calculus and differential equations, this resource will provide a
thorough and evolving foundation for using mapping diagrams on the
pathway to higher mathematics.