Questions: Draw a graph to match the description given. H(x) is decreasing over (-∞, ∞), but the derivative does not exist at x=-4.

Solution Steps

The function $H(x)$ is decreasing over $(-\infty, \infty)$, which means its derivative $H'(x) < 0$ for all $x$ except at $x = -4$, where the derivative does not exist. A common function that is decreasing everywhere except at a point where the derivative does not exist is a piecewise function with a cusp or a corner at $x = -4$.

A possible function that satisfies these conditions is: \[ H(x) = \begin{cases} -x - 4 & \text{if } x < -4 \\ -x + 4 & \text{if } x \geq -4 \end{cases} \] This function is linear and decreasing everywhere, but it has a corner at $x = -4$ where the derivative does not exist.

Final Answer