Question

type the correct answer in each box. use numerals instead of words. if necessary, use / for the fraction bar(s)
the graph represents the piecewise function
$f(x)=\

$$\begin{cases}\\square, \\text{ if } -3 \\leq x < -1 \\\\\\square, \\text{ if } -1 \\leq x \\leq 1\\end{cases}$$

$

Explanation:

Step1: Analyze the first piece ($-3 \leq x < -1$)

The graph from $x = -3$ to $x = -1$ is a line. Let's find two points on this line. When $x = -3$, $f(x) = 0$; when $x = -1$, $f(x) = 2$ (wait, no, wait. Wait, looking at the graph, from $x=-3$ (closed dot) to $x=-1$ (open dot? Wait, no, the graph: at $x=-3$, it's a closed dot, then goes to $x=-1$ (open dot?) Wait, no, the graph shows from $x=-3$ (closed) to $x=-1$ (open?) Wait, no, the line from $(-3, 0)$ to $(-1, 2)$? Wait, no, wait the coordinates: when $x=-3$, $y=0$; when $x=-1$, $y=2$. So the slope $m = \frac{2 - 0}{-1 - (-3)} = \frac{2}{2} = 1$. So the equation is $y - 0 = 1 \times (x - (-3))$, so $y = x + 3$. Let's check: when $x=-3$, $y=0$ (correct). When $x=-1$, $y = -1 + 3 = 2$ (but the dot at $x=-1$ for this piece is open? Wait, no, the piece is $-3 \leq x < -1$, so at $x=-1$, it's not included. The next piece is from $-1 \leq x \leq 1$, which is a horizontal line? Wait, no, the graph from $x=-1$ to $x=1$: the closed dot at $x=-1$ (y=5?) Wait, no, the graph: looking at the grid, from $x=-1$ to $x=1$, the line is horizontal at $y=5$? Wait, no, the top part: at $x=-1$ to $x=1$, the y-value is 5? Wait, the graph has a horizontal segment at y=5 from x=-1 to x=1 (closed dots at both ends). And the segment from x=-3 to x=-1 is a line from (-3, 0) to (-1, 2). Wait, let's re-examine:

Wait, the graph:

- For $-3 \leq x < -1$: the line connects (-3, 0) to (-1, 2). So slope is (2 - 0)/(-1 - (-3)) = 2/2 = 1. So equation is $y = x + 3$. Let's verify: when x=-3, y=0 (correct). When x=-1, y=2 (but since x < -1, the endpoint at x=-1 is open, so that's okay).

- For $-1 \leq x \leq 1$: the graph is a horizontal line at y=5 (since from x=-1 to x=1, the y-value is 5, with closed dots at both ends).

Wait, maybe I misread the graph. Let's look again:

The graph:

- Left part: from x=-6 (or left) to x=-3, horizontal at y=0 (closed dot at x=-3).

- Then from x=-3 (closed) to x=-1 (open), a line going up to (x=-1, y=2)? Wait, no, the dot at x=-1 for that segment is open, and then from x=-1 (closed) to x=1 (closed), horizontal at y=5.

Wait, the problem's piecewise function is:

$f(x) = \begin{cases} \text{[first function]} & \text{if } -3 \leq x < -1 \\ \text{[second function]} & \text{if } -1 \leq x \leq 1 \end{cases}$

So first, for $-3 \leq x < -1$:

Points: (-3, 0) and (-1, 2) (but x=-1 is not included here). So slope $m = (2 - 0)/(-1 - (-3)) = 2/2 = 1$. So equation: $y = x + 3$. Let's check x=-3: y=0 (correct). x=-2: y=-2 + 3 = 1 (which is on the line).

For $-1 \leq x \leq 1$: the graph is horizontal at y=5 (since from x=-1 to x=1, the y-value is 5, with closed dots at both ends). So the function here is $f(x) = 5$.

Wait, but let's confirm:

At x=-1, the first piece ends (open dot at y=2), and the second piece starts with a closed dot at y=5? Wait, maybe I misread the y-axis. Let's check the y-axis: the top dot is at y=9? No, the grid lines: the horizontal line from x=-1 to x=1 is at y=5 (since between y=4 and y=6, the middle is 5). Yes, the segment from x=-1 to x=1 is horizontal at y=5.

So:

For $-3 \leq x < -1$: the line from (-3, 0) to (-1, 2) has equation $y = x + 3$.

For $-1 \leq x \leq 1$: the horizontal line is $y = 5$.

Wait, but let's check the first piece again. When x=-3, y=0 (correct, closed dot). When x=-1, y=2 (but the dot at x=-1 for the first piece is open, and the second piece has a closed dot at x=-1, y=5. So that's correct.

So the piecewise function is:

$f(x) = \begin{cases} x + 3, & \text{if } -3 \leq x < -1 \\ 5, & \text{if } -1 \leq x \leq 1 \end{cases}$

Step2: Confirm the first piece

The line from (-3, 0) to (-1, 2): slope 1, y-intercept 3. So equation is $f(x) = x + 3$ for $-3 \leq x < -1$.

Step3: Confirm the second piece

The horizontal line from x=-1 (closed dot) to x=1 (closed dot) at y=5, so $f(x) = 5$ for $-1 \leq x \leq 1$.

Answer:

For the first box (when $-3 \leq x < -1$): $x + 3$

For the second box (when $-1 \leq x \leq 1$): $5$

So the piecewise function is:

$f(x) =

$$\begin{cases} \boldsymbol{x + 3}, & \text{if } -3 \leq x < -1 \\ \boldsymbol{5}, & \text{if } -1 \leq x \leq 1 \end{cases}$$

$