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 $arcsin = \sin^{-1}$.

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.