Exploring the definition of a function in math and how it can inform my understanding of functions in programming

0.no_inputs.php

<?php
/**
 * This function has no inputs, so it is not a function from a mathematical
 * perspective.  Furthermore, the function is not predictable because it depends on 
 * the state of the underlying program
 */
function say_hello() {
  
  $name = 'World';

  /**
   * Note we are using external state to alter our function's behavior. This logic may
   * be better placed elsewhere in order to keep the function more consistent and predictable
   */
  if( user_is_logged_in_right_now() ) {
    $user = get_logged_in_user();
    $name = $user->first_name;
  }
  
  return "Hello, {$name}";
}

1.more_predictable_hello.php

<?php
/**
 * This function achieves the same result in a more predictable way and does 
 * qualify as a function in a mathematical sense
 */
function say_hello_to( $name ) {
  return "Hello, {$name}";
}

2.key_value_store.php

<?php
$my_array = [
	'foo' => 'bar',
	'baz' => 'bat',
];

3.key_value_store_2.php

<?php
/**
 * For the input (i.e. key) 'baz', this changes the output (i.e. value) from 'bat' to 'bae'
 */
$my_array['baz'] = 'bae';

4.key_value_store_3.php

<?php
/**
 * Again, we have 'bae' replacing 'bat' as the value associated with the key 'bae'
 */
$my_array = [
	'foo' => 'bar',
	'baz' => 'bat',
	'baz' => 'bae',
];

5.key_value_store_4.php

<?php
$my_array = [
	'foo' => 'bar',
	'baz' => [ 'bat', 'bae' ]
];