Question

on a piece of paper, draw the graph of a continuous function $y = f(x)$ that satisfies the following three conditions.
$f(x)<0$ for $x < - 4$,
$f(x)<0$ for $-4 < x < 4$,
$f(x)<0$ for $4 < x$
approximate your function by picking a segment from the following for each of the sections of your graph, first for $x < -4$, then for $-4 < x < 4$, and then for $4 < x$. (you should, of course, imagine sliding the pieces vertically up or down to make the function you create be continuous.)
for $x < -4$, use segment
for $-4 < x < 4$, use segment
for $4 < x$, use segment

Explanation:

Step1: Recall derivative - function relationship

If \(f^{\prime}(x)<0\) on an interval, the function \(y = f(x)\) is decreasing on that interval.

Step2: Analyze the intervals

We have three intervals \(x < - 4\), \(-4

Step3: Choose segments

Since the function is decreasing in each interval, we need to pick segments that are decreasing - sloping downwards from left - to - right for each of the given intervals. Then, we can vertically adjust these segments to make the function continuous.

Answer:

Pick decreasing - sloping segments for each of the intervals \(x < - 4\), \(-4