1. The problem asks us to fill in the function table for the function $$y = -\ln(x) - 1$$ at specific values of $$x$$ between 1 and 4, divided into 5 equal intervals. 2. First, find the interval step size. Since the interval is from 1 to 4 and there are 5 boxes, the step size is $$\frac{4-1}{5} = 0.6$$. 3. The $$x$$ values for the boxes are: - Box 1: $$1 + 0.6 = 1.6$$ - Box 2: $$1 + 2 \times 0.6 = 2.2$$ - Box 3: $$1 + 3 \times 0.6 = 2.8$$ - Box 4: $$1 + 4 \times 0.6 = 3.4$$ - Box 5: $$4$$ (given) 4. Calculate the corresponding $$y$$ values using $$y = -\ln(x) - 1$$: - Box 6 (at $$x=1$$): $$y = -\ln(1) - 1 = -0 - 1 = -1$$ (given) - Box 7 (at $$x=1.6$$): $$y = -\ln(1.6) - 1 \approx -0.470 - 1 = -1.470$$ - Box 8 (at $$x=2.2$$): $$y = -\ln(2.2) - 1 \approx -0.788 - 1 = -1.788$$ - Box 9 (at $$x=2.8$$): $$y = -\ln(2.8) - 1 \approx -1.030 - 1 = -2.030$$ - Box 10 (at $$x=3.4$$): $$y = -\ln(3.4) - 1 \approx -1.223 - 1 = -2.223$$ - Box 11 (at $$x=4$$): $$y = -\ln(4) - 1 \approx -1.386 - 1 = -2.386$$ 5. Final answers rounded to 3 decimal places: - Box 1 = 1.6 - Box 2 = 2.2 - Box 3 = 2.8 - Box 4 = 3.4 - Box 5 = 4 - Box 6 = -1 - Box 7 = -1.470 - Box 8 = -1.788 - Box 9 = -2.030 - Box 10 = -2.223 - Box 11 = -2.386 These values fill the function table for the given function and interval.