Question

directions: drag the domain interval to the correct expression that matches the given piecewise graph.

$f(x)=$

expression	domain interval
$2$
$-\frac{3}{4}x + 5\frac{5}{8}$

(the graph shows distance (miles) on the y - axis from 0 to 3 and time (hours) on the x - axis from 0 to 8. there are domain interval options: $0\geq x>2$, $0\leq x<1$, $0\leq x<2$, $1\leq x<6$, $1\geq x>6$, $0\geq x>1$, $0\leq x\leq7.5$, $0\geq x\geq7.5$, $6\leq x\leq7.5$, $6\geq x\geq7.5$)

Explanation:

To solve this, we analyze the piecewise function's graph and match each expression to its domain interval by examining the slope and behavior of each segment:

Step 1: Analyze \( f(x) = 2x \)

The first segment of the graph (from \( x = 0 \) to \( x = 1 \)) is a line passing through the origin with a slope of 2 (since \( 2(0) = 0 \) and \( 2(1) = 2 \), matching the graph’s start). The domain here is \( 0 \leq x < 1 \) (since at \( x = 1 \), the graph switches to a horizontal line).

Step 2: Analyze \( f(x) = 2 \)

The middle segment is a horizontal line at \( y = 2 \), spanning from \( x = 1 \) to \( x = 6 \) (inclusive at \( x = 1 \), exclusive at \( x = 6 \), as the graph remains constant here). Thus, the domain is \( 1 \leq x < 6 \).

Step 3: Analyze \( f(x) = -\frac{3}{4}x + 5\frac{5}{8} \)

The third segment is a line with a negative slope (decreasing) starting at \( x = 6 \) and ending at \( x = 7.5 \) (since at \( x = 6 \), \( -\frac{3}{4}(6) + 5\frac{5}{8} = -\frac{18}{4} + \frac{45}{8} = -\frac{36}{8} + \frac{45}{8} = \frac{9}{8} \)? Wait, no—wait, the graph ends at \( y = 0 \) when \( x = 7.5 \) (since \( -\frac{3}{4}(7.5) + 5\frac{5}{8} = -\frac{22.5}{4} + \frac{45}{8} = -\frac{45}{8} + \frac{45}{8} = 0 \)). So the domain for this segment is \( 6 \leq x \leq 7.5 \).

Final Matches:

• \( 2x \): Domain interval \( \boldsymbol{0 \leq x < 1} \)
• \( 2 \): Domain interval \( \boldsymbol{1 \leq x < 6} \)
• \( -\frac{3}{4}x + 5\frac{5}{8} \): Domain interval \( \boldsymbol{6 \leq x \leq 7.5} \)

Filling the Table:

Expression	Domain Interval
\( 2 \)	\( 1 \leq x < 6 \)
\( -\frac{3}{4}x + 5\frac{5}{8} \)	\( 6 \leq x \leq 7.5 \)

(Note: The domain intervals are selected from the given options to match the graph’s segments.)

Answer:

To solve this, we analyze the piecewise function's graph and match each expression to its domain interval by examining the slope and behavior of each segment:

Step 1: Analyze \( f(x) = 2x \)

The first segment of the graph (from \( x = 0 \) to \( x = 1 \)) is a line passing through the origin with a slope of 2 (since \( 2(0) = 0 \) and \( 2(1) = 2 \), matching the graph’s start). The domain here is \( 0 \leq x < 1 \) (since at \( x = 1 \), the graph switches to a horizontal line).

Step 2: Analyze \( f(x) = 2 \)

The middle segment is a horizontal line at \( y = 2 \), spanning from \( x = 1 \) to \( x = 6 \) (inclusive at \( x = 1 \), exclusive at \( x = 6 \), as the graph remains constant here). Thus, the domain is \( 1 \leq x < 6 \).

Step 3: Analyze \( f(x) = -\frac{3}{4}x + 5\frac{5}{8} \)

The third segment is a line with a negative slope (decreasing) starting at \( x = 6 \) and ending at \( x = 7.5 \) (since at \( x = 6 \), \( -\frac{3}{4}(6) + 5\frac{5}{8} = -\frac{18}{4} + \frac{45}{8} = -\frac{36}{8} + \frac{45}{8} = \frac{9}{8} \)? Wait, no—wait, the graph ends at \( y = 0 \) when \( x = 7.5 \) (since \( -\frac{3}{4}(7.5) + 5\frac{5}{8} = -\frac{22.5}{4} + \frac{45}{8} = -\frac{45}{8} + \frac{45}{8} = 0 \)). So the domain for this segment is \( 6 \leq x \leq 7.5 \).

Final Matches:

• \( 2x \): Domain interval \( \boldsymbol{0 \leq x < 1} \)
• \( 2 \): Domain interval \( \boldsymbol{1 \leq x < 6} \)
• \( -\frac{3}{4}x + 5\frac{5}{8} \): Domain interval \( \boldsymbol{6 \leq x \leq 7.5} \)

Filling the Table:

Expression	Domain Interval
\( 2 \)	\( 1 \leq x < 6 \)
\( -\frac{3}{4}x + 5\frac{5}{8} \)	\( 6 \leq x \leq 7.5 \)

(Note: The domain intervals are selected from the given options to match the graph’s segments.)