I am looking to get a little group of php functions working which i need on the whole page. I am including them during my index.php primary template via require_once "my_functions.php".

They operate in the index.php template that's what's promising.

There exists a wordpress plugin that let us us write php within our content products using the following syntax: {source}<?php echo "hello world!"; ?>{/source}

However, when attempting to make use of the functions defined during my_funcitons.php, php thinks they are not available and creates a fatal error ... undefined method or function.

Any ideas?

I do think you need to write a module, implement your require_once within it and insert the created module in articles.

Avoid creating an excessive amount of php code in WYSIWYG editors, you are mixing quite happy with programming logic, beating CMS's purpose: manage content. And when you've non-technical people behind the body this could be a real headache later on.

Use Direct PHP.