How to make enemy chase for longer distances than it's initial "notice" range

I would create an "isFollowing" bool; and then use it as below. I also took the liberty to create separate functions for the different behaviours in Update(), this is what I would do but your preference may differ.


public class WastrelChase : MonoBehaviour
{
 public Transform player;
 private Animator anim;
 private Rigidbody myRigidbody;
 private Vector3 moveDirection;
 public CapsuleCollider cc;
 public BoxCollider pc;
 public Slider healthbar;
 public Collider Attack;
 public GameObject hurt;
 public GameObject lunge;
 public GameObject growl;
 public GameObject shriek;
 private float triggerDist = 5;
 private float maxDist = 20;
 private bool isFollowing = false;

 void OnTriggerEnter(Collider attack)
 {
 if (attack.gameObject.tag == "PlayerAttack") {

 healthbar.value -= 15;
 anim.SetBool("IsInjured", true);
 }


 }

 // Use this for initialization
 void Start()
 {
 anim = GetComponent();
 myRigidbody = GetComponent();
 cc.enabled = false;
 pc.enabled = false;
 transform.GetComponent().constraints = RigidbodyConstraints.FreezePositionY | RigidbodyConstraints.FreezeRotationX | RigidbodyConstraints.FreezeRotationY | RigidbodyConstraints.FreezeRotationZ;
 }

 // Update is called once per frame
 void Update()
 {
 moveDirection = Vector3.zero;

 if (!isFollowing) 
 {
 NotFollowing ();

 if (Vector3.Distance (player.position, this.transform.position) < triggerDist) 
 {
 isFollowing = true;
 }
 }
 else 
 {
 Following ();

 if (Vector3.Distance (player.position, this.transform.position) > maxDist) 
 {
 isFollowing = false;
 }
 }

 CheckHealth ();
 }

 private void NotFollowing()
 {
 anim.SetBool("IsIdle", true);
 anim.SetBool("IsWalking", false);
 anim.SetBool("IsStriking", false);
 }

 private void Following()
 {

 Vector3 direction = player.position - this.transform.position;
 direction.y = 0;

 this.transform.rotation = Quaternion.Slerp(this.transform.rotation,
 Quaternion.LookRotation(direction), 0.1f);

 anim.SetBool("IsIdle", false);
 if (direction.magnitude > .8)
 {
 moveDirection = direction.normalized;
 anim.SetBool("IsWalking", true);
 anim.SetBool("IsStriking", false);
 anim.SetBool("IsInjured", false);
 }
 else
 {
 anim.SetBool("IsStriking", true);
 anim.SetBool("IsWalking", false);
 }


 }

 private void CheckHealth()
 {
 if (healthbar.value <= 0){
 anim.SetBool("IsVulnerable", true);
 }
 else
 {
 anim.SetBool("IsVulnerable", false);
 }
 }
}


So i would recommend looking into a finite state machine to achieve the simple behaviour that you're looking for, where the idle, chase, and seek are all separate behaviours.

With that you can trigger a chase state, which will directly follow the player for that minimum view threshold, and once out of that state, move to a Seek state, which would move towards a direction for X amount of time, until either that enemy has either noticed the player, and has reentered a chase state, or has lost the player, and will revert to its idle state.

Unity does have an example of this, but there loads of resources online.