Let’s break the question into parts and analyze step by step:

1. Functions Overview:

   • $f(x) = 4x + 3$: This is a linear function.
   • $g(x)$: From the table, the values of $y$ (dependent on $x$) are provided:
     $(x, y) = (0, 24), (1, 20), (2, 16), (3, 12), (4, 9), (5, 8), (6, 4)$.
     Notice that $g(x)$ is decreasing non-linearly.

2. Average Rate of Change: The average rate of change of a function between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $\text{Average Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1}.$

3. First Blank: Interval where the average rate of change of $g(x)$ is greater than $f(x)$: For $f(x)$, the slope (rate of change) is constant and equal to $4$ (the coefficient of $x$).
   Calculate the average rate of change of $g(x)$ over intervals and compare with $4$:

   • From $x = 0$ to $x = 1$:
     $\text{Rate of Change of } g = \frac{20 - 24}{1 - 0} = -4.$
   • From $x = 1$ to $x = 2$:
     $\text{Rate of Change of } g = \frac{16 - 20}{2 - 1} = -4.$
   • This shows that the rate of change of $g(x)$ is consistently less than $f(x)$.

   Answer: The average rate of change of $g(x)$ is not greater than $f(x)$ on any interval provided.

4. Second Blank: As $x$ increases, the average rates of change of $f$ and $g$:

   • $f(x)$ has a constant rate of change ($4$).
   • $g(x)$ has a decreasing rate of change (as shown above, rates are negative and stay constant).

   Answer: The average rates of change of $f$ and $g$ remain constant, respectively.

5. Third Blank: When $x = 7$, the value of $g(x)$:

   • Using the trend of $g(x)$, the table stops at $x = 6$, where $g(6) = 4$.
   • Assuming $g(x)$ continues its trend, you can extrapolate to find $g(7)$:
     From $x = 5$ to $x = 6$, $g(x)$ decreases by $-4$. Thus, $g(7)$ would decrease by another $-4$: $g(7) = 4 - 4 = 0.$

   Answer: When $x = 7$, $g(x) = 0$.

6. General Statement: An exponential function (e.g., doubling growth) will eventually exceed any linear function because exponential growth accelerates at an increasing rate.

---

Final Fill-in-the-Blanks:

1. Over the interval none, the average rate of change of $g$ is greater than the average rate of change of $f$.
2. As the value of $x$ increases, the average rates of change of $f$ and $g$ remain constant, respectively.
3. When the value of $x = 7$, the value of $g(x)$ is 0.
4. It can be further generalized that a quantity increasing exponentially will exceed a quantity increasing linearly.

---

Would you like more details on the calculations or explanations?

Related Questions:

1. How do you compute the average rate of change for exponential functions?
2. What are the differences between linear and exponential growth models?
3. How does the derivative represent the rate of change for a function?
4. How do we predict trends beyond the given data in a function?
5. How can we compare rates of change using graphs?

Tip:

Always check the behavior of the function (linear, quadratic, exponential, etc.) before making predictions about its growth or decline.