Perlin Noise Problem with Falloff

Question

I know this might be a bit of a noob question, but I can't figure out what I'm doing wrong, and the math that I have done should make this work correctly. I am trying to make a perlin noise map that is normal until a specific range, at which point it would decrease to where all the values trend towards zero, creating an island in an ocean. I am currently generating perlin "sample" values, and then dividing them by a factor which stays close to one up to a specific bound, and then increases polynomialy. The problem however, is that the noise stays stuck in the top right corner, and is darker than I would expect. Here is the mathematical function I am using graphed on Desmos where n is the boundary and x is the distance from the center of the map:

And here is the main body code I am using to generate the noise map:

using System.Collections;
using System.Collections.Generic;
using UnityEngine;

public class GenNoise : MonoBehaviour {

public int width = 256;
public int height = 256;

public float scale = 20f;

public float offsetX;
public float offsetY;

public float falloffRange = 100;
public float falloffPower = 10;

public void RenderNoiseMap()
{
    Renderer renderer = GetComponent<Renderer>();
    renderer.sharedMaterial.mainTexture = GenerateTexture();
}

Texture2D GenerateTexture () {
    Texture2D texture = new Texture2D(width, height);

    for (int x = 0; x < width; x++)
    {
        for (int y = 0; y < height; y++)
        {
            Color color = GenerateColor(x, y);
            texture.SetPixel(x, y, color);
        }
    }

    texture.Apply();
    return texture;
}

Color GenerateColor(int x, int y)
{
    float xCoord = ((float)x / width * scale) + offsetX;
    float yCoord = ((float)y / height * scale) + offsetY;
    float falloffFactor = FalloffFactor(x, y, width, height);

    float sample = Mathf.PerlinNoise(xCoord, yCoord);
    sample = sample / falloffFactor;
    return new Color(sample, sample, sample);
}

float FalloffFactor(int x, int y, int width, int height)
{
    float falloffFactor = 1;
    float centerX = (float)width / 2;
    float centerY = (float)height / 2;

    float distanceFromCenter = Mathf.Sqrt(Mathf.Pow(2, ((float)x - centerX)) + Mathf.Pow(2, ((float)y - centerY)));
    falloffFactor = Mathf.Pow(2, (Mathf.Pow(falloffPower, (distanceFromCenter / falloffRange)) + 1));

    return falloffFactor;
}
}

And Here is the code that I am using to generate the image in the editor:

using UnityEngine;
using UnityEditor;

[CustomEditor(typeof(GenNoise))]
public class NewBehaviourScript : Editor {
public override void OnInspectorGUI()
{
    DrawDefaultInspector();

    GenNoise script = (GenNoise)target;
    if (GUILayout.Button("Generate"))
    {
        script.RenderNoiseMap();
    }
}

}

Here is a picture of the result:

Answer 1

I think your Mathf.pow functions all have the arguments reversed. e.g. Mathf.Pow(2, ((float)y - centerY))) is 2^(y-centerY), where you probably meant to do (y-centerY)^2 considering that you're trying to calculate a distance.