Question

complete the table using the graph below. draw (graph of a piecewise function on a coordinate plane with grid lines, showing different segments with open and closed circles at various x - values like - 3, 1, 3 etc. and y - values like 1, - 1, 3 etc.) show your work here

Explanation:

Since the table to be completed is not shown, we can analyze the graph to find the function's values at different \( x \)-values. Let's assume the table has columns for \( x \) and \( f(x) \), and we'll find values for key \( x \)-points (like \( x = -3, x = 1, x = 3 \), etc.) by examining the graph:

Step 1: Analyze \( x = -3 \)

• At \( x = -3 \), there are two points: a closed dot (filled circle) at \( y = 3.5 \) (or \( \frac{7}{2} \)) and an open dot (hollow circle) at \( y = 1 \).
• Closed dots mean the function includes that point, open dots mean it doesn’t. So \( f(-3) = 3.5 \) (or \( \frac{7}{2} \)).

Step 2: Analyze \( x \) between \( -3 \) and \( 1 \) (excluding endpoints)

• The horizontal line with open dots at \( x = -3 \) (open) and \( x = 1 \) (open) has \( y = 1 \). For \( -3 < x < 1 \), the function follows this line, so \( f(x) = 1 \) here.

Step 3: Analyze \( x = 1 \)

• At \( x = 1 \), there’s an open dot at \( y = 1 \) and a closed dot at \( y = -1 \). So \( f(1) = -1 \) (closed dot is included).

Step 4: Analyze \( x \) between \( 1 \) and \( 3 \) (excluding endpoints)

• The horizontal line with closed dots at \( x = 1 \) and \( x = 3 \) has \( y = -1 \). So for \( 1 < x < 3 \), \( f(x) = -1 \).

Step 5: Analyze \( x = 3 \)

• At \( x = 3 \), there’s an open dot at \( y = 3 \) and a closed dot at \( y = -1 \). So \( f(3) = -1 \) (closed dot is included), and the line with the open dot at \( x = 3 \) starts at \( y = 3 \) for \( x > 3 \).

Step 6: Analyze \( x > 3 \)

• For \( x > 3 \), the line with the open dot at \( x = 3 \) ( \( y = 3 \)) has a slope. But since we need to complete the table, we focus on key \( x \)-values. For \( x = 3 \), \( f(3) = -1 \) (from the horizontal line), and for \( x > 3 \), \( f(x) \) follows the line with \( y = 3 \) at \( x = 3 \) (open dot) and decreases.

If the table includes \( x = -3, 1, 3 \), the values would be:

\( x \)	\( f(x) \)
\( 0 \) (between \( -3 \) and \( 1 \))	\( 1 \)
\( 1 \)	\( -1 \)
\( 2 \) (between \( 1 \) and \( 3 \))	\( -1 \)
\( 3 \)	\( -1 \)
\( 4 \) (greater than \( 3 \))	\( 2.5 \) (or \( \frac{5}{2} \)) (from the decreasing line)

To fully complete the table, we need the exact columns/rows of the table. If you provide the table’s structure (e.g., \( x \)-values to evaluate), we can refine the answer.

For example, if the table asks for \( f(-3) \), \( f(0) \), \( f(1) \), \( f(2) \), \( f(3) \):

\( x \)	\( f(x) \)
\( 0 \)	\( 1 \)
\( 1 \)	\( -1 \)
\( 2 \)	\( -1 \)
\( 3 \)	\( -1 \)

(Note: The exact \( y \)-values depend on the grid’s precision; adjust if the graph’s grid has different intervals.)

Answer:

Since the table to be completed is not shown, we can analyze the graph to find the function's values at different \( x \)-values. Let's assume the table has columns for \( x \) and \( f(x) \), and we'll find values for key \( x \)-points (like \( x = -3, x = 1, x = 3 \), etc.) by examining the graph:

Step 1: Analyze \( x = -3 \)

• At \( x = -3 \), there are two points: a closed dot (filled circle) at \( y = 3.5 \) (or \( \frac{7}{2} \)) and an open dot (hollow circle) at \( y = 1 \).
• Closed dots mean the function includes that point, open dots mean it doesn’t. So \( f(-3) = 3.5 \) (or \( \frac{7}{2} \)).

Step 2: Analyze \( x \) between \( -3 \) and \( 1 \) (excluding endpoints)

• The horizontal line with open dots at \( x = -3 \) (open) and \( x = 1 \) (open) has \( y = 1 \). For \( -3 < x < 1 \), the function follows this line, so \( f(x) = 1 \) here.

Step 3: Analyze \( x = 1 \)

• At \( x = 1 \), there’s an open dot at \( y = 1 \) and a closed dot at \( y = -1 \). So \( f(1) = -1 \) (closed dot is included).

Step 4: Analyze \( x \) between \( 1 \) and \( 3 \) (excluding endpoints)

• The horizontal line with closed dots at \( x = 1 \) and \( x = 3 \) has \( y = -1 \). So for \( 1 < x < 3 \), \( f(x) = -1 \).

Step 5: Analyze \( x = 3 \)

• At \( x = 3 \), there’s an open dot at \( y = 3 \) and a closed dot at \( y = -1 \). So \( f(3) = -1 \) (closed dot is included), and the line with the open dot at \( x = 3 \) starts at \( y = 3 \) for \( x > 3 \).

Step 6: Analyze \( x > 3 \)

• For \( x > 3 \), the line with the open dot at \( x = 3 \) ( \( y = 3 \)) has a slope. But since we need to complete the table, we focus on key \( x \)-values. For \( x = 3 \), \( f(3) = -1 \) (from the horizontal line), and for \( x > 3 \), \( f(x) \) follows the line with \( y = 3 \) at \( x = 3 \) (open dot) and decreases.

If the table includes \( x = -3, 1, 3 \), the values would be:

\( x \)	\( f(x) \)
\( 0 \) (between \( -3 \) and \( 1 \))	\( 1 \)
\( 1 \)	\( -1 \)
\( 2 \) (between \( 1 \) and \( 3 \))	\( -1 \)
\( 3 \)	\( -1 \)
\( 4 \) (greater than \( 3 \))	\( 2.5 \) (or \( \frac{5}{2} \)) (from the decreasing line)

To fully complete the table, we need the exact columns/rows of the table. If you provide the table’s structure (e.g., \( x \)-values to evaluate), we can refine the answer.

For example, if the table asks for \( f(-3) \), \( f(0) \), \( f(1) \), \( f(2) \), \( f(3) \):

\( x \)	\( f(x) \)
\( 0 \)	\( 1 \)
\( 1 \)	\( -1 \)
\( 2 \)	\( -1 \)
\( 3 \)	\( -1 \)

(Note: The exact \( y \)-values depend on the grid’s precision; adjust if the graph’s grid has different intervals.)