91Ó°ÊÓ

Let's create a simple piecewise function for the given intervals. We can use linear functions to represent the decreasing and increasing behavior of the function for different intervals. Define the piecewise function as: f(x) = \[ \begin{cases} -x - 1, & \text{if } x \in [-2,1] \\ x - 1, & \text{if } x \in [1,5] \end{cases} \] This piecewise function represents a function that is decreasing on the interval [-2,1] and increasing on the interval [1,5]. Now, let's graph this function.

To graph the piecewise function, we'll sketch the corresponding linear functions on their respective intervals. 1. For x in [-2,1], f(x) = -x - 1: - When x=-2, f(x) = 1. This point is (-2, 1). - When x=1, f(x) = -2. This point is (1, -2). 2. For x in [1,5], f(x) = x - 1: - When x=1, f(x) = 0. This point is (1, 0). - When x=5, f(x) = 4. This point is (5, 4). Now, plot the piecewise function using these points: - For the interval [-2,1]: - Plot a line segment connecting the points (-2, 1) and (1, -2). Since f(1) is not included this interval, we should indicate an open circle at the point (1, -2). - For the interval [1,5]: - Plot a line segment connecting the points (1, 0) and (5, 4). Since f(1) is included in this interval, we should indicate a solid circle at the point (1, 0). With these line segments and points plotted, you have successfully graphed a function that is decreasing on the interval [-2,1] and increasing on the interval [1,5].