1. We are asked to find the values of $g(0)$, $g(2)$, and $g(4)$ for the function $g(x) = -5x$. 2. To find $g(0)$, substitute $x=0$ into the function: $$g(0) = -5 \times 0 = 0$$ 3. To find $g(2)$, substitute $x=2$: $$g(2) = -5 \times 2 = -10$$ 4. To find $g(4)$, substitute $x=4$: $$g(4) = -5 \times 4 = -20$$ 5. Next, we calculate the average rate of change $\frac{\Delta g(x)}{\Delta x}$ over the intervals $[0,2]$, $[0,4]$, and $[2,4]$. 6. The average rate of change formula is: $$\frac{\Delta g(x)}{\Delta x} = \frac{g(x_2) - g(x_1)}{x_2 - x_1}$$ 7. For the interval $[0,2]$: $$\frac{\Delta g(x)}{\Delta x} = \frac{g(2) - g(0)}{2 - 0} = \frac{-10 - 0}{2} = \frac{-10}{2} = -5$$ 8. For the interval $[0,4]$: $$\frac{\Delta g(x)}{\Delta x} = \frac{g(4) - g(0)}{4 - 0} = \frac{-20 - 0}{4} = \frac{-20}{4} = -5$$ 9. For the interval $[2,4]$: $$\frac{\Delta g(x)}{\Delta x} = \frac{g(4) - g(2)}{4 - 2} = \frac{-20 - (-10)}{2} = \frac{-20 + 10}{2} = \frac{-10}{2} = -5$$ 10. Notice that the average rate of change is constant and equals $-5$, which matches the slope of the linear function $g(x) = -5x$. Final answers: - $g(0) = 0$ - $g(2) = -10$ - $g(4) = -20$ - Average rate of change over $[0,2]$, $[0,4]$, and $[2,4]$ is $-5$ in each case.