Questions: Complete the description of the piecewise function graphed below. 1 if -6 ≤ x ≤ -1 f(x)= if -1 < x ≤ 3 ↑ if 3 < x ≤ 6

Complete the description of the piecewise function graphed below.

Find the equation of the line segment for \(-6 \leq x \leq -1\).

The line segment passes through the points \((-6, -6)\) and \((-1, 4)\). The slope is \(\frac{4 - (-6)}{-1 - (-6)} = \frac{10}{5} = 2\). Using the point-slope form with the point \((-1, 4)\), we get \(y - 4 = 2(x - (-1))\), so \(y - 4 = 2(x + 1)\), or \(y = 2x + 2 + 4 = 2x + 6\). Thus, \(f(x) = 2x + 6\) for \(-6 \leq x \leq -1\).

Find the equation of the line segment for \(-1 < x \leq 3\).

The line segment is horizontal and passes through the point \((3, 2)\). The equation of a horizontal line is of the form \(y = c\), where \(c\) is a constant. In this case, \(y = 2\). Thus, \(f(x) = 2\) for \(-1 < x \leq 3\).

Find the equation of the line segment for \(3 < x \leq 6\).

The line segment passes through the points \((3, 2)\) and \((6, -3)\). The slope is \(\frac{-3 - 2}{6 - 3} = \frac{-5}{3}\). Using the point-slope form with the point \((3, 2)\), we get \(y - 2 = -\frac{5}{3}(x - 3)\), so \(y - 2 = -\frac{5}{3}x + 5\), or \(y = -\frac{5}{3}x + 7\). Thus, \(f(x) = -\frac{5}{3}x + 7\) for \(3 < x \leq 6\).

\( f(x) = \begin{cases} 2x + 6, & -6 \leq x \leq -1 \\ 2, & -1 < x \leq 3 \\ -\frac{5}{3}x + 7, & 3 < x \leq 6 \end{cases} \)

\( f(x) = \begin{cases} 2x + 6, & -6 \leq x \leq -1 \\ 2, & -1 < x \leq 3 \\ -\frac{5}{3}x + 7, & 3 < x \leq 6 \end{cases} \)