Override a function with child themes functions.php - wordpress

Question

As the title reads I'm trying to modify a function called by a parent theme in my child, I know that the child theme is set to be loaded beforehand so I'm curious if this is even possible?
My parent theme has a function called ajax_search_box() that I'd like to modify a query in, and I'd rather not modify the parent theme files in case I need to update it down the road.. what would be the best way to do this?
Also, for bonus points, how would I go about doing this with a widget as well? Thanks in advance!
function SearchFilter($query) {
// If 's' request variable is set but empty
if (isset($_GET['s']) && empty($_GET['s']) && $query->is_main_query()){
$query->is_search = true;
$query->is_home = false;
}
return $query;
}
add_filter('pre_get_posts','SearchFilter');
function ajax_search_box() {
if (isset($_GET["q"]))
{
$q = $_GET["q"];
global $wpdb;
$q = mysql_real_escape_string($q);
$q = $wpdb->escape($q);
$query = array(
'post_status' => 'publish',
'order' => 'DESC',
's' => $q
);
$get_posts = new WP_Query;
$posts = $get_posts->query( $query );
// Check if any posts were found.
if ( ! $get_posts->post_count )
die();
//Create an array with the results
foreach ( $posts as $post )
echo $post->post_title . "|" . $post->ID . "\n";
}
die();
}
// creating Ajax call for WordPress
add_action( 'wp_ajax_nopriv_ajax_search_box', 'ajax_search_box' );
add_action( 'wp_ajax_ajax_search_box', 'ajax_search_box' );

Answer 1

the parent theme needs to check if(function_exists('ajax_search_box')) and if it doesn't exist then it will declare it.
If the parent theme checks to see if the function exists, then you can declare it first and have it do what you want.
If the parent theme does not check, get in touch with the theme author to see if they will throw that change in for the next update....and code it yourself too. That way when the theme updates then you will still be good to go.

Answer 2

Break free of functions.php, write your own plugin.
<?php
/**
* Plugin Name: Manipulate the Parent
* Requires: PHP5.3+
*/
add_action( 'after_setup_theme', function()
{
remove_filter( 'pre_get_posts','SearchFilter' );
// now add your own filter
add_filter( 'pre_get_posts', 'your_callback_for_your_filter' );
});
function your_callback_for_your_filter()
{
// do stuff
}