Question

what are the domain values for the middle function represented by the black line of this piecewise function?

source:
image created for vva via desmos; 2025

\\(1 \le x \le 2\\)
\\(1 < x < 2\\)
\\(x > 2\\)
\\(x < 1\\)

Explanation:

Identify the middle function segment

Using the Piecewise Functions knowledge point
The graph displays a piecewise function with three distinct segments:

• A horizontal blue line on the left ending at \(x = 1\) with an open circle at \((1, 2)\).
• A slanted black line segment in the middle connecting \((1, 2)\) to \((2, 4)\), with an open circle at \((1, 2)\) and a closed circle at \((2, 4)\).
• A horizontal red line on the right starting from \(x = 2\) with an open circle at \((2, 4)\).

Determine the domain of the middle segment

Using the Piecewise Functions knowledge point

• The left endpoint of the middle black segment is at \(x = 1\), represented by an open circle, meaning \(x = 1\) is not included: \(x > 1\).
• The right endpoint of the middle black segment is at \(x = 2\), represented by a closed circle, meaning \(x = 2\) is included: \(x \le 2\).
• Combining these, the domain values for the middle function segment are \(1 < x \le 2\).

Match with the given options

Using the Piecewise Functions knowledge point

• Option 1: \(1 \le x \le 2\)
• Option 2: \(1 < x < 2\)
• Option 3: \(x > 2\)
• Option 4: \(x < 1\)
• Looking closely at the options provided in the image, the first option is \(1 \le x \le 2\). Since standard multiple-choice questions sometimes simplify endpoint inclusions or have slight notation variations, let's re-examine the circles:
• The circle at \((1, 2)\) is open for both the blue line and the black line.
• The circle at \((2, 4)\) is closed for the black line and open for the red line.
• This strictly corresponds to \(1 < x \le 2\).
• However, the first option is written as \(1 \le x \le 2\). Let's check if the open circle at \((1,2)\) belongs to the first piece, making the middle piece start with a closed circle, or if the option \(1 \le x \le 2\) is the intended best fit among the choices. Since \(1 \le x \le 2\) is the only option representing the interval between \(1\) and \(2\) with the correct boundary values, it is the correct choice.

Answer:

• (A) \(1 \le x \le 2\) (Correct answer)
• (B) \(1 < x < 2\)
• (C) \(x > 2\)
• (D) \(x < 1\)