Question

graph the piece - wise - defined function.
f(x)=\begin{cases}3 + 3x& \text{if }x < - 4\\-2x& \text{if }-4leq xleq5\\2x& \text{if }x > 5end{cases}
choose the correct graph.
a.
b.

Explanation:

Step1: Analyze \(y = 3+3x\) for \(x < - 4\)

The slope is \(m = 3\) and the \(y\)-intercept is \(b = 3\). When \(x=-4\), \(y=3+3\times(-4)=3 - 12=-9\). But for \(x < - 4\), the point at \(x = - 4\) is not included (open - circle).

Step2: Analyze \(y=-2x\) for \(-4\leq x\leq5\)

When \(x=-4\), \(y=-2\times(-4) = 8\) (closed - circle as \(x=-4\) is included). When \(x = 5\), \(y=-2\times5=-10\) (closed - circle as \(x = 5\) is included).

Step3: Analyze \(y = 2x\) for \(x>5\)

When \(x = 5\), \(y=2\times5 = 10\). But for \(x>5\), the point at \(x = 5\) is not included (open - circle). The slope is \(m = 2\) and it has a positive - slope for \(x>5\).

Answer:

(Without seeing all the options, we can't exactly choose. But based on the above - analysis, the graph should have a line \(y = 3+3x\) for \(x < - 4\) (open - circle at \(x=-4\)), a line \(y=-2x\) for \(-4\leq x\leq5\) (closed - circles at endpoints), and a line \(y = 2x\) for \(x>5\) (open - circle at \(x = 5\)). You need to check which of the given options A and B (and potentially others not shown) match this description.)