Question

graph the piece - wise - defined function. f(x)=\

$$\begin{cases}3 + 3x&\\text{if }x < - 4\\\\-2x&\\text{if }-4\\leq x\\leq5\\\\2x&\\text{if }x > 5\\end{cases}$$

choose the correct graph. a. image of graph a b. image of graph b c. image of graph c d. image of graph d

Explanation:

Step1: Analyze first - part of function

For \(f(x)=3 + 3x\) when \(x\lt - 4\). When \(x=-4\), \(f(-4)=3+3\times(-4)=3 - 12=-9\). The slope of \(y = 3+3x\) is \(m = 3\) and it is a line for \(x\lt - 4\), so it has an open - circle at \(x=-4\).

Step2: Analyze second - part of function

For \(f(x)=-2x\) when \(-4\leq x\leq5\). When \(x=-4\), \(f(-4)=-2\times(-4) = 8\) (closed - circle since \(-4\) is included in this interval). When \(x = 5\), \(f(5)=-2\times5=-10\) (closed - circle since \(5\) is included). The slope of \(y=-2x\) is \(-2\).

Step3: Analyze third - part of function

For \(f(x)=2x\) when \(x\gt5\). When \(x = 5\), \(f(5)\) from this part is not defined, but as \(x\) approaches \(5\) from the right, \(y = 2x\) has an open - circle at \(x = 5\) and the slope is \(m = 2\).

Answer:

Without seeing the actual graphs A, B, C, D, you would look for a graph that has an open - circle at \(x=-4\) for the line \(y = 3+3x\), a line segment with closed - circles at \(x=-4\) and \(x = 5\) for \(y=-2x\), and an open - circle at \(x = 5\) for the line \(y = 2x\).