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
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.