Question

f(x)=\begin{cases}3x + 2&\text{if }x<5\\-3x - 2&\text{if }xgeq5end{cases}

Explanation:

Step1: Analyze the function for \(x < 5\)

The function \(y = 3x+2\) for \(x < 5\) is a linear - function with slope \(m = 3\) and \(y\) - intercept \(b = 2\). When \(x = 5\), \(y=3\times5 + 2=15 + 2=17\). Since \(x<5\), the point \((5,17)\) is not included in this part of the graph, so we have an open - circle at \((5,17)\) for \(y = 3x + 2\).

Step2: Analyze the function for \(x\geq5\)

The function \(y=-3x - 2\) for \(x\geq5\) is a linear function with slope \(m=-3\) and when \(x = 5\), \(y=-3\times5-2=-15 - 2=-17\). Since \(x\geq5\), the point \((5, - 17)\) is included in this part of the graph, so we have a closed - circle at \((5,-17)\) for \(y=-3x - 2\).

Step3: Check the graphs

For the function \(y = 3x+2\) (\(x < 5\)), as \(x\) increases, \(y\) increases. For the function \(y=-3x - 2\) (\(x\geq5\)), as \(x\) increases, \(y\) decreases.

Answer:

The correct graph is the one that has an open - circle at the point \((5,17)\) for the line \(y = 3x+2\) (for \(x < 5\)) and a closed - circle at the point \((5,-17)\) for the line \(y=-3x - 2\) (for \(x\geq5\)). Without seeing the exact details of each graph labeled a, b, c, and d, you can identify the correct one based on these characteristics. If we assume the graphs are labeled in a standard way, you should look for a graph where the line \(y = 3x+2\) (with an open - circle at \(x = 5\)) and the line \(y=-3x - 2\) (with a closed - circle at \(x = 5\)) are drawn correctly.