1. **State the problem:** We are given the function $f(x) = 5x + 4$ and two points $x_0 = 3$ and $x_1 = 7$. We want to find the average rate of change of the function between these two points using the Leibniz technique (difference quotient). 2. **Formula used:** The average rate of change of a function $f(x)$ between $x_0$ and $x_1$ is given by: $$\frac{f(x_1) - f(x_0)}{x_1 - x_0}$$ This represents the slope of the secant line connecting the points $(x_0, f(x_0))$ and $(x_1, f(x_1))$ on the graph of $f$. 3. **Calculate $f(x_0)$ and $f(x_1)$:** $$f(3) = 5(3) + 4 = 15 + 4 = 19$$ $$f(7) = 5(7) + 4 = 35 + 4 = 39$$ 4. **Apply the formula:** $$\frac{f(7) - f(3)}{7 - 3} = \frac{39 - 19}{4} = \frac{20}{4} = 5$$ 5. **Interpretation:** The average rate of change of $f(x)$ between $x=3$ and $x=7$ is 5. Since $f(x)$ is a linear function with slope 5, this matches the constant rate of change. **Final answer:** The average rate of change of $f(x)$ from $x=3$ to $x=7$ is $5$.