Question

answer attempt 2 out of 2
$f(x) = \

$$\begin{cases} -2x - 4 & \\text{for } -5 < x \\leq 0 \\\\ \\frac{5}{2}x - 8 & \\text{for } 0 < x \\leq 6 \\end{cases}$$

$
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Explanation:

Step1: Analyze the first piece

For the interval \(-5 < x \leq 0\), the function is \(y=-2x - 4\). Let's check the left - hand part of the graph. At \(x = 0\), \(y=-2(0)-4=-4\), which matches the filled dot at \((0, - 4)\). The slope of \(y=-2x - 4\) is \(-2\) (negative slope), and the line from \(x=-5\) (open circle) to \(x = 0\) (filled circle) has a negative slope, which is consistent.

Step2: Analyze the second piece

For the interval \(0 < x\leq6\), the function is \(y=\frac{5}{2}x-8\). The slope of this line is \(\frac{5}{2}\) (positive slope). At \(x = 0\), \(y=\frac{5}{2}(0)-8=-8\) (open circle at \((0, - 8)\) as per the graph). At \(x = 6\), \(y=\frac{5}{2}(6)-8=15 - 8 = 7\), which matches the filled dot at \((6,7)\). Also, when \(y = 0\), \(\frac{5}{2}x-8=0\Rightarrow\frac{5}{2}x=8\Rightarrow x=\frac{16}{5}=3.2\), which is consistent with the graph's intersection with the \(x\) - axis around \(x = 3\) (more precisely \(x=\frac{16}{5}\)).

Answer:

The piece - wise function \(f(x)=

$$\begin{cases}-2x - 4&\text{for }-5 < x\leq0\\\frac{5}{2}x-8&\text{for }0 < x\leq6\end{cases}$$

\) correctly represents the given graph.