Question

h(x) = \

$$\begin{cases} 2 - x &, -7 \\leq x \\leq 5 \\\\ 3x - 21 &, 5 < x \\leq 9 \\end{cases}$$

what is the graph of h?
choose 1 answer:
a (graph with points and lines)
b (graph with points and lines)

Explanation:

Step1: Analyze the first piece \(y = 2 - x\) for \(-7\leq x\leq5\)

• Find the endpoints:
• When \(x=-7\), \(y = 2-(-7)=9\). So the left - endpoint is \((-7,9)\) (closed circle since \(x = - 7\) is included).
• When \(x = 5\), \(y=2 - 5=-3\). So the right - endpoint of the first segment is \((5,-3)\) (closed circle since \(x = 5\) is included in the first interval). The slope of \(y = 2 - x\) is \(- 1\), so the line is decreasing.

Step2: Analyze the second piece \(y=3x - 21\) for \(5\lt x\leq9\)

• Find the endpoints:
• When \(x = 5\) (not included), \(y=3\times5-21=15 - 21=-6\). So there is an open circle at \((5,-6)\).
• When \(x = 9\), \(y=3\times9-21=27 - 21 = 6\). So the right - endpoint is \((9,6)\) (closed circle since \(x = 9\) is included). The slope of \(y = 3x-21\) is \(3\), so the line is increasing.
• Now, check the options:
• In option A, the first segment has a left - endpoint at \((-7,9)\) (closed), right - endpoint at \((5,-3)\) (closed), and the second segment has an open circle at \((5,-6)\) and a closed circle at \((9,6)\) with a positive slope.
• In option B, the right - endpoint of the first segment is incorrect (it should be \((5,-3)\) but in option B's first segment, the right - endpoint seems to be at a different \(y\) - value) and the open circle at \(x = 5\) for the second segment is not at \((5,-6)\) as required.

Answer:

A