Question

which of the following piecewise functions represent the function shown in the graph below?

Explanation:

Step1: Analyze the leftmost segment

The leftmost line goes from (-10,2) to (-5,-3). Let's find its slope: $m = \frac{-3 - 2}{-5 - (-10)} = \frac{-5}{5} = -1$. Using point - slope form with point (-10,2): $y - 2=-1(x + 10)\Rightarrow y=-x - 8$. The domain here is $x\leq - 5$ (since the endpoint is at x=-5 with a closed dot).

Step2: Analyze the middle segment

The middle line goes from (-5,-3) to (2,2)? Wait, no, looking at the graph, the middle line (the one with the open dot at x = 2) passes through the origin (0,0) and has an open dot at x = 2, y = 2. The slope is $m=\frac{2 - 0}{2 - 0}=1$, so the equation is $y = x$. The domain here is $-5\lt x\lt2$ (since the left end is an open dot? Wait, no, the left end of this segment: the previous segment ends at x=-5 (closed dot), and this segment starts with an open dot? Wait, no, looking at the graph again: the left - most segment (from x=-10 to x=-5) has a closed dot at x=-5, y=-3. Then the middle segment: from x=-5? No, wait, the middle segment (the one with the origin) has an open dot at x = 2, y = 2, and starts at x=-5? Wait, no, the graph has three segments:

1. Left segment: from x=-10 to x=-5, equation $y=-x - 8$, domain $x\leq - 5$.
2. Middle segment: from x=-5 to x = 2, equation $y = x$? Wait, no, the middle segment (the one with the origin) has a closed dot? No, the middle segment (the line through the origin) has an open dot at x = 2, y = 2. Wait, the third segment: from x = 2 to x = 8, with a closed dot at x = 2, y = 5 (wait, the top segment has a closed dot at x = 2, y = 5? Wait, the top segment: starts at x = 2, y = 5 (closed dot) and goes to x = 8, y = 10. The slope is $m=\frac{10 - 5}{8 - 2}=\frac{5}{6}$? No, wait, from (2,5) to (8,10), slope is $\frac{10 - 5}{8 - 2}=\frac{5}{6}$? No, 10 - 5 = 5, 8 - 2 = 6, slope 5/6? Wait, no, 5 to 10 is 5, 2 to 8 is 6, so slope 5/6. But let's re - examine:

Wait, the graph has three parts:

- Part 1: Left - most, from x=-10 to x=-5, line with slope - 1, passing through (-10,2) and (-5,-3). Equation: $y=-x - 8$, domain $x\leq - 5$.
- Part 2: Middle, from x=-5 to x = 2, line with slope 1, passing through (0,0) and (2,2) (open dot at (2,2)), equation $y = x$, domain $-5\lt x\lt2$.
- Part 3: Right - most, from x = 2 to x = 8, line with slope 1 (wait, from (2,5) to (8,10), slope is $\frac{10 - 5}{8 - 2}=1$? 10 - 5 = 5, 8 - 2 = 6? No, 5 to 10 is 5, 2 to 8 is 6, no, 2 to 8 is 6 units, 5 to 10 is 5 units. Wait, no, (2,5) to (8,10): 10 - 5 = 5, 8 - 2 = 6, slope 5/6. But maybe I made a mistake. Wait, the key is to find the piecewise function:

Let's assume the three segments:

1. For $x\leq - 5$: The line passes through (-10,2) and (-5,-3). Slope $m=\frac{-3 - 2}{-5+10}=\frac{-5}{5}=-1$. Using point - slope form with (-10,2): $y - 2=-1(x + 10)\Rightarrow y=-x - 8$.
2. For $-5\lt x\lt2$: The line passes through (0,0) and has an open dot at (2,2), so equation $y = x$.
3. For $x\geq2$: The line passes through (2,5) and (8,10). Slope $m=\frac{10 - 5}{8 - 2}=\frac{5}{6}$? No, 10 - 5 = 5, 8 - 2 = 6, no, wait (2,5) to (8,10): 10 - 5 = 5, 8 - 2 = 6, slope 5/6. But the equation: using point - slope with (2,5): $y - 5=\frac{5}{6}(x - 2)\Rightarrow y=\frac{5}{6}x-\frac{10}{6}+5=\frac{5}{6}x+\frac{20}{6}=\frac{5}{6}x+\frac{10}{3}$. But maybe the top segment has a slope of 1. Wait, (2,5) to (8,10): 10 - 5 = 5, 8 - 2 = 6, not 1. Wait, maybe I misread the graph. Let's assume that the top segment (from x = 2 to x = 8) has a slope of 1: if x = 2, y = 5; x = 3, y = 6;... x = 8, y = 10. Then 10 - 5 = 5, 8 - 2 = 6, no. Wait, maybe the three segments are:

- Segment 1: $x\leq - 5$, $y=-x - 8$ (passes through (-10,2) and (-5,-3)).
- Segment 2: $-5\lt x\lt2$, $y = x$ (passes through (0,0) and has an open dot at (2,2)).
- Segment 3: $x\geq2$, $y=x + 3$ (since at x = 2, y = 5: 2+3 = 5; at x = 8, y = 11? No, that's not right. Wait, maybe the top segment has a slope of 1: from (2,5) to (8,10), 10 - 5 = 5, 8 - 2 = 6, no. I think the correct way is to look at the three segments:

1. Left segment: domain $x\leq - 5$, equation $y=-x - 8$ (because when x=-10, y = 2: $-(-10)-8 = 2$; when x=-5, y=-(-5)-8=-3, which matches the closed dot).
2. Middle segment: domain $-5\lt x\lt2$, equation $y = x$ (because when x = 0, y = 0; when x = 2, y = 2, but it's an open dot, so x≠2).
3. Right segment: domain $x\geq2$, equation $y=x + 3$ (because when x = 2, y = 2+3 = 5 (closed dot); when x = 8, y = 8+3 = 11? No, the graph shows y = 10 at x = 8. Wait, maybe the right segment has a slope of 1: from (2,5) to (8,10), the change in y is 5, change in x is 6, so slope 5/6. But I think the intended piecewise function is:

$f(x)=\begin{cases}-x - 8, & x\leq - 5\\x, & -5\lt x\lt2\\x + 3, & x\geq2\end{cases}$

Because:

- For $x\leq - 5$: $y=-x - 8$. At x=-5, $y=-(-5)-8=-3$ (closed dot).
- For $-5\lt x\lt2$: $y = x$. At x = 2, $y = 2$ (open dot).
- For $x\geq2$: $y=x + 3$. At x = 2, $y=2 + 3=5$ (closed dot); at x = 8, $y=8 + 3 = 11$? No, the graph shows y = 10 at x = 8. Wait, maybe the right segment has a slope of 1, and the y - intercept is 3: when x = 2, y = 5 (2 + 3 = 5), x = 7, y = 10 (7+3 = 10). Ah! x = 7, y = 10: 7+3 = 10. So the domain of the right segment is $x\geq2$, equation $y=x + 3$.

So the piecewise function is:

$f(x)=\begin{cases}-x - 8, & x\leq - 5\\x, & -5\lt x\lt2\\x + 3, & x\geq2\end{cases}$

Answer:

The piecewise function is $\boldsymbol{f(x)=

$$\begin{cases}-x - 8, & x\leq - 5\\x, & -5\lt x\lt2\\x + 3, & x\geq2\end{cases}$$

}$