Functions and MySQL Connectivity issues

I am trying to create a script (from a tutorial I try to adapt to my own needs) and I am running into a few problems. Here I go...

I have an /includes folder which contains the following: database.php and functions.php with the following content:

database.php

<?php

// Database connectivity stuff

$host     = "localhost"; // Hostname for the database. Usually localhost
$username = "root"; // Username used to connect to the database
$password = "root"; // Password for the username used to connect to the database
$database = "blog"; // The database used

// Connect to the database using mysqli_connect
$connection = mysqli_connect($host, $username, $password, $database);

// Check the connection for errors
	if (mysqli_connect_errno($connection)) {
	// Stop the whole page from loading if errors occur
	die("<br />Could not connect to the database. Please check the settings and try again.") . mysqli_connect_error() . mysqli_connect_errno();
}


?>

functions.php

<?php

// Functions file for the system
function show_posts($user_id) {
    $posts = array();
    $sql = "SELECT body, stamp from posts where user_id = '$user_id' order by stamp desc";
    $result = mysqli_query($connection, $sql);

    while ($data = mysqli_fetch_assoc($result)) {
    	$posts = array(
	    		'stamp' => $data->stamp,
	    		'user_id' => $user_id,
	    		'body' => $data->body
    		);
    }
    return $posts;
}

function show_users() {
	$users = array();
	$sql = "SELECT id, username FROM users WHERE status = 'active' ORDER BY username";
	$result = mysqli_query($connection, $sql);

	while ($data = mysqli_fetch_array($result)) {
		$users[$data->id] = $data->username;
	}
	return $users;
}

function following($user_id) {
	$users = array();
	$sql = "SELECT DISTINCT user_id FROM following WHERE follower_id = $user_id";
	$result = mysqli_query($connection, $sql);

	while ($data = mysqli_fetch_assoc($result)) {
		array_push($users, $data->user_id);
	}
	return $users;
}

?>

And here's the code that I run to display the users in users.php

      <?php

        $users = show_users();
        foreach ($users as $key => $value) {
          echo $key . " " . $value;
        }

      ?>

That throws me the following errors:

Notice: Undefined variable: connection in /Applications/MAMP/htdocs/blog/includes/functions.php on line 22

Warning: mysqli_query() expects parameter 1 to be mysqli, null given in /Applications/MAMP/htdocs/blog/includes/functions.php on line 22

Warning: mysqli_fetch_array() expects parameter 1 to be mysqli_result, null given in /Applications/MAMP/htdocs/blog/includes/functions.php on line 24

I've tried to pass the $connection value to the show_users(); function with no results... any help would be appreciated. I am able to display the users from the database if I introduce the following code in users.php, but I want to try and keep things as clear as possible, without cluttering the files unnecessary.

       <?php 
        $users = array();
        $query = "SELECT id, username FROM users WHERE status = 'active' ORDER BY username";
        $result = mysqli_query($connection, $query);
        if ($result) {
        while ($member = mysqli_fetch_assoc($result)) { 
          echo strip_tags($member['username'])  . "<br />";
        }
        }
        else {
          echo "There are no posts to display. Why not add a new one?";
        }
        mysqli_free_result($result);
        mysqli_close($connection);
      ?>

PS: The users.php file also has require('includes/database.php'); and require('includes/functions.php'). I guess I'm trying to recreate the last bit of code into a working function... and I can't seem to do it.  Any help is appreciated. I hope it makes an sense...

-- Andrei