To convert the given natural language description into functional Gurobi code that solves the optimization problem, we first need to identify and formalize the objective function and the constraints.

The objective function aims to maximize the value of `1` times the quantity of bananas plus `8` times the quantity of granola bars. This can be represented as:

Maximize: `1 * bananas + 8 * granola_bars`

Given variables:
- `bananas`: Quantity of bananas, which can be a non-whole number.
- `granola_bars`: Quantity of granola bars, which must be a whole number.

Constraints based on the resources/attributes:
1. Dollar cost constraint: The total cost should be at least $38 and at most $111.
   - Cost per banana: $14
   - Cost per granola bar: $8
2. Healthiness rating constraint: The total healthiness rating must be greater than or equal to 145 and less than or equal to 255.
   - Healthiness rating per banana: 11
   - Healthiness rating per granola bar: 9
3. Additional linear constraint: `-4 * bananas + 9 * granola_bars >= 0`

Let's represent the problem in Gurobi code, considering all constraints and the objective function:

```python
from gurobipy import *

# Create a model
m = Model("Optimization_Problem")

# Define variables
bananas = m.addVar(vtype=GRB.CONTINUOUS, name="bananas")
granola_bars = m.addVar(vtype=GRB.INTEGER, name="granola_bars")

# Objective function: Maximize 1 * bananas + 8 * granola_bars
m.setObjective(1 * bananas + 8 * granola_bars, GRB.MAXIMIZE)

# Constraints
# 1. Dollar cost constraint
m.addConstr(14 * bananas + 8 * granola_bars >= 38, name="min_cost")
m.addConstr(14 * bananas + 8 * granola_bars <= 111, name="max_cost")

# 2. Healthiness rating constraint
m.addConstr(11 * bananas + 9 * granola_bars >= 145, name="min_health")
m.addConstr(11 * bananas + 9 * granola_bars <= 255, name="max_health")

# 3. Additional linear constraint
m.addConstr(-4 * bananas + 9 * granola_bars >= 0, name="additional_constraint")

# Optimize the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print("Bananas:", bananas.x)
    print("Granola Bars:", granola_bars.x)
    print("Objective Value:", m.objVal)
else:
    print("No optimal solution found")
```