Can You Graph an Increasing and Decreasing Function with These Points?

To determine whether it is possible to create an increasing and decreasing function using the two given points, we need to consider the concept of piecewise functions, which allow different behavior in different intervals.

Step 1: Recognize the nature of the problem. We have two points, which often prompts the consideration of either a linear or piecewise linear function.

Step 2: Analyze the end behavior. With the two points, we can begin by assuming a segment, a line, or a curve that connects them. To fulfill the request for both increasing and decreasing properties, the simplest approach involves using a piecewise-defined function. This can be achieved by defining one segment that is increasing and another that is decreasing, or vice versa.

Step 3: Implement a piecewise approach. Consider a three-point scenario by hypothetically placing a midpoint between the two existing points. This hypothetical point allows for an increase from the first point to the midpoint and a decrease from the midpoint to the second point. Such a function will satisfy the condition of having both increasing and decreasing segments.

Thus, it is indeed possible to define a piecewise function that achieves increasing behavior over one part and decreasing behavior over another by simply using the points or introducing an intermediate point to toggle between increasing and decreasing segments.

Therefore, the solution to the problem is Possible.