1. The problem gives two functions: $$p_N(x) = -x + 13.2$$ and $$K(x) = 0.1x^3 - 0.9x^2 + 3x + 16$$. 2. We want to understand or analyze these functions. Let's start by stating what each function represents: - $$p_N(x)$$ is a linear function. - $$K(x)$$ is a cubic polynomial. 3. For the linear function $$p_N(x) = -x + 13.2$$: - The slope is $$-1$$. - The y-intercept is $$13.2$$. 4. For the cubic function $$K(x) = 0.1x^3 - 0.9x^2 + 3x + 16$$: - It has a leading term $$0.1x^3$$ which dominates for large $$|x|$$. - The other terms affect the shape and turning points. 5. If the goal is to find intersections or analyze behavior, we can set $$p_N(x) = K(x)$$: $$-x + 13.2 = 0.1x^3 - 0.9x^2 + 3x + 16$$ 6. Rearranging all terms to one side: $$0.1x^3 - 0.9x^2 + 3x + 16 + x - 13.2 = 0$$ $$0.1x^3 - 0.9x^2 + 4x + 2.8 = 0$$ 7. This cubic equation can be solved for $$x$$ to find intersection points. Since the user did not specify a particular question, this is the analysis of the first function given.