Question

question graph the following function on the axes provided.
$f(x)=\

$$\begin{cases}2x + 5&\\text{for }x < - 3\\\\-5x-4&\\text{for }x>0\\end{cases}$$

$ click and drag to make a line. click the line to delete it. click on an endpoint to change it.

Explanation:

Step1: Analyze the first - part of the piece - wise function

For \(x < - 3\), the function is \(y = 2x+5\). When \(x=-3\), \(y = 2\times(-3)+5=-6 + 5=-1\). Since \(x < - 3\), we have an open - circle at the point \((-3,-1)\) and the slope of the line is \(m = 2\) and the \(y\) - intercept for this part is \(b = 5\). We can find another point, for example, when \(x=-4\), \(y=2\times(-4)+5=-8 + 5=-3\).

Step2: Analyze the second - part of the piece - wise function

For \(-3\leqslant x\leqslant0\), the function is \(y=-5x - 4\). When \(x=-3\), \(y=-5\times(-3)-4 = 15-4 = 11\). When \(x = 0\), \(y=-5\times0-4=-4\). We have a closed - circle at \((-3,11)\) and a closed - circle at \((0,-4)\) and the slope of the line is \(m=-5\) and the \(y\) - intercept for this part is \(b=-4\).

Step3: Analyze the third - part of the piece - wise function

For \(x>0\), the function is \(y = 0\). We have an open - circle at the point \((0,0)\) and it is a horizontal line \(y = 0\) for \(x>0\).

Answer:

Graph the three lines as described above: an open - circle at \((-3,-1)\) with the line \(y = 2x + 5\) for \(x < - 3\), a line segment with closed - circles at \((-3,11)\) and \((0,-4)\) for \(-3\leqslant x\leqslant0\) using \(y=-5x - 4\), and an open - circle at \((0,0)\) with the horizontal line \(y = 0\) for \(x>0\).