Question

answer attempt 1 out of 3
$f(x) = \

$$\begin{cases} \\hphantom{\ ule{1.5em}{0ex}} \\text{for} \\hphantom{\ ule{1.5em}{0ex}} \\\\ \\hphantom{\ ule{1.5em}{0ex}} \\text{for} \\hphantom{\ ule{1.5em}{0ex}} \\end{cases}$$

$
< > ≤ ≥ = ≠ add rule remove rule submit answer

Explanation:

Step1: Analyze the left line

The left line passes through points \((-4, 0)\) and \((0, -4)\) (and the closed dot at \((1, -5)\)? Wait, no, let's check the slope. Wait, the left line: let's take two points. When \(x = -4\), \(y = 0\); when \(x = 0\), \(y = -4\)? Wait, no, the line goes from the left, let's find the slope. Let's take \((-4, 0)\) and \((1, -5)\)? Wait, the closed dot is at \((1, -5)\)? Wait, the left line: let's calculate the slope between \((-4, 0)\) and \((0, -4)\). Slope \(m=\frac{-4 - 0}{0 - (-4)}=\frac{-4}{4}=-1\). So the equation is \(y - 0=-1(x + 4)\), so \(y=-x - 4\). Now, the domain for this part: the left line goes until \(x\leq1\) (since the closed dot is at \(x = 1\), \(y=-5\); let's check: when \(x = 1\), \(y=-1 - 4=-5\), which matches the closed dot. So first piece: \(f(x)=-x - 4\) for \(x\leq1\).

Step2: Analyze the right line

The right line has an open dot at \((5, -9)\) and goes to \((10, 1)\)? Wait, no, the right line: let's find two points. The open dot is at \((5, -9)\) and the end at \((10, 1)\)? Wait, slope between \((5, -9)\) and \((10, 1)\): \(m=\frac{1 - (-9)}{10 - 5}=\frac{10}{5}=2\). Let's find the equation. Using point - slope form with \((10, 1)\): \(y - 1 = 2(x - 10)\), \(y=2x-20 + 1=2x - 19\). Wait, let's check at \(x = 5\): \(y=2(5)-19=10 - 19=-9\), which matches the open dot. So the domain for this part is \(x>5\) (since the open dot is at \(x = 5\)). Wait, but wait, between \(x = 1\) and \(x = 5\), is there a function? Wait, the graph: the left line ends at \(x = 1\) (closed dot), then there's a gap until \(x = 5\) (open dot), then the right line. Wait, maybe I misread. Wait, the left line: let's re - examine. The left line: when \(x=-4\), \(y = 0\); when \(x = 1\), \(y=-5\) (closed dot). So slope is \(\frac{-5 - 0}{1 - (-4)}=\frac{-5}{5}=-1\), so equation \(y=-x - 4\), which is correct. Then the right line: open dot at \((5, -9)\), and goes to \((10, 1)\). Slope is \(\frac{1-(-9)}{10 - 5}=2\), equation \(y=2x - 19\) (as above). So the piecewise function:

First piece: \(f(x)=-x - 4\) for \(x\leq1\)

Second piece: Wait, no, maybe there's a middle part? Wait, the graph: the left line (with closed dot at \(x = 1\)), then a gap, then the right line (open dot at \(x = 5\)). Wait, maybe the second piece is from \(x>5\)? Wait, let's check the right line. The right line starts at the open dot at \((5, -9)\) and goes to \((10, 1)\). So the equation is \(y = 2x-19\) for \(x>5\). Wait, but maybe I made a mistake. Let's re - check the left line. Let's take \(x=-4\), \(y = 0\): \(y=-(-4)-4=0\), correct. \(x = 1\): \(y=-1 - 4=-5\), correct (closed dot). Then the right line: at \(x = 5\), open dot, \(y=-9\): \(2(5)-19=10 - 19=-9\), correct. At \(x = 10\), \(y=2(10)-19=1\), correct (the arrow at \(x = 10\), \(y = 1\)). So the piecewise function is:

\(f(x)=\begin{cases}-x - 4 & \text{for } x\leq1\\2x - 19 & \text{for } x>5\end{cases}\)

Wait, but maybe there's a third piece? But the problem shows two boxes, so maybe two pieces. Let's confirm the domains. The left part: \(x\leq1\) (closed dot at \(x = 1\)), the right part: \(x>5\) (open dot at \(x = 5\)).

Step1 (revised): Left piece equation and domain

We find two points on the left line: \((-4, 0)\) and \((1, -5)\). The slope \(m=\frac{-5 - 0}{1 - (-4)}=-1\). Using point - slope form with \((-4, 0)\): \(y-0=-1(x + 4)\), so \(y=-x - 4\). The domain for this part is \(x\leq1\) (since the closed dot is at \(x = 1\)).

Step2 (revised): Right piece equation and domain

For the right line, we take two points: \((5, -9)\) (open dot) and \((10, 1)\). The slope \(m=\frac{1 - (-9)}{10 - 5}=2\). Using point - slope form with \((10, 1)\): \(y - 1=2(x - 10)\), so \(y=2x-19\). The domain for this part is \(x>5\) (since the open dot is at \(x = 5\)).

Answer:

\(f(x)=

$$\begin{cases}-x - 4 & \text{for } x\leq1\\2x - 19 & \text{for } x>5\end{cases}$$

\)