Your second function overwrites the query of the first. It’s not necessary anyway as you can add as many post types to the $query->set line in the first function as you need, like this:
$query->set( 'post_type', array( 'post', 'first_function', 'my-second-post-type','another-post-type' ) );
BTW, is your post type actually called ‘first_function’? Just asking as that’s the name of the function too.
Hope that helps