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A graph on a coordinate plane shows a function with one line segment and one curved segment that are disjoint.The coordinate plane has an x-axis range of negative 10 to 10 and a y-axis range of negative 10 to 10. There is a line segment connecting an open point at left-parenthesis negative 6 comma negative 3 right-parenthesis and a closed point at left-parenthesis 2 comma 2 right-parenthesis. A curve connects an open point at left-parenthesis 1 comma 7 right-parenthesis and a closed point at left-parenthesis 5 comma 3 right-parenthesis.

Question
Use the graph of the function to answer the question.
Find the domain and range of the function shown in the graph. Write the domain and range using interval notation.
(1 point)
Responses

domain: left-bracket negative 3 comma 2 right-parenthesis union left-parenthesis 3 comma 7 right-bracket
range: left-bracket negative 6 comma negative 1 right-parenthesis union left-bracket 1 comma 5 right-parenthesis
domain: Image with alt text: left-bracket negative 3 comma 2 right-parenthesis union left-parenthesis 3 comma 7 right-bracket range: Image with alt text: left-bracket negative 6 comma negative 1 right-parenthesis union left-bracket 1 comma 5 right-parenthesis

domain: left-parenthesis negative 6 comma negative 1 right-bracket union left-parenthesis 1 comma 5 right-bracket
range: left-parenthesis negative 3 comma 2 right-bracket union left-bracket 3 comma 7 right-parenthesis
domain: Image with alt text: left-parenthesis negative 6 comma negative 1 right-bracket union left-parenthesis 1 comma 5 right-bracket range: Image with alt text: left-parenthesis negative 3 comma 2 right-bracket union left-bracket 3 comma 7 right-parenthesis

domain: left-parenthesis negative 3 comma 2 right-bracket union left-bracket 3 comma 7 right-parenthesis
range: left-parenthesis negative 6 comma negative 1 right-bracket union left-parenthesis 1 comma 5 right-bracket
domain: Image with alt text: left-parenthesis negative 3 comma 2 right-bracket union left-bracket 3 comma 7 right-parenthesis range: Image with alt text: left-parenthesis negative 6 comma negative 1 right-bracket union left-parenthesis 1 comma 5 right-bracket

domain: left-bracket negative 6 comma negative 1 right-parenthesis union left-bracket 1 comma 5 right-parenthesis
range: left-bracket negative 3 comma 2 right-parenthesis union left-parenthesis 3 comma 7 right-bracket

1 answer

To find the domain and range of the given function, we analyze the specified points of the line segment and the curve segment.

Line Segment:

The line segment connects:

  • Open point: \((-6, -3)\)
  • Closed point: \((2, 2)\)

From \((-6, -3)\) to \((2, 2)\), the domain includes all \(x\)-values in this range:

  • Domain for line segment: \[ (-6, 2] \]

Curve Segment:

The curve connects:

  • Open point: \((1, 7)\)
  • Closed point: \((5, 3)\)

From \((1, 7)\) to \((5, 3)\), the domain includes all \(x\)-values in this range:

  • Domain for curve segment: \[ (1, 5] \]

Combined Domain:

Combining the two segments, the complete domain is: \[ (-6, 2] \cup (1, 5] \]

Range:

Next, we determine the range from both segments.

  1. Line Segment:

    • \(y\) values range from the lowest point of \(-3\) at \(x=-6\) (open) to \(2\) at \(x=2\) (closed).
    • Range for line segment: \[ [-3, 2] \]

  2. Curve Segment:

    • The \(y\) values range from \(7\) at \(x=1\) (open) down to \(3\) at \(x=5\) (closed).
    • Range for curve segment: \[ (3, 7] \]

Combined Range:

Combining both components, the complete range is: \[ [-3, 2] \cup (3, 7] \]

Final Answer:

The domain and range of the function in interval notation are:

  • Domain: \[ (-6, 2] \cup (1, 5] \]

  • Range: \[ [-3, 2] \cup (3, 7] \]

Given the provided responses, the correct domain and range based on our calculations do not exactly match any of the listed responses. However, the closest correct form of domain and range according to analysis is:

Domain: \( (-6, 2] \cup (1, 5] \)
Range: \( [-3, 2] \cup (3, 7] \)