Part of a line is graphed at (2, 1) and (6, -3). Which inequalities could describe the domain and range?

a. Domain:\ 2<x<6 Range:\ 2<y<6
b. Domain:\ 2<x\le6 Range:\ -3\le y<1
c. Domain:\ 2\le x<6\ \ Range:\ y<1
d. Domain:\ x\ge2 Range:\ -3<y\le1

1 answer

To determine which inequalities describe the domain and range of the line segment defined by the points (2, 1) and (6, -3), we need to analyze the coordinates of these points.

  1. Domain: The x-coordinates of the points are 2 and 6. Therefore, the domain of the line segment includes all x-values from 2 to 6, which can be expressed as:

    • \(2 \leq x \leq 6\) or \(2 < x < 6\)
  2. Range: The y-coordinates of the points are 1 and -3. Because the y-values go from 1 to -3, the range of the line segment includes all y-values from -3 (the lowest) to 1 (the highest). This can be expressed as:

    • \(-3 \leq y \leq 1\) or \(-3 < y < 1\)

Analyzing the given options:

a. Domain: \(2 < x < 6\) Range: \(2 < y < 6\)

  • Incorrect. The range does not include values above 1.

b. Domain: \(2 < x \le 6\) Range: \(-3 \leq y < 1\)

  • Correct. Covers the appropriate domain and range values.

c. Domain: \(2 \le x < 6\) Range: \(y < 1\)

  • Incorrect. The range should include the lower bound of -3.

d. Domain: \(x \ge 2\) Range: \(-3 < y \le 1\)

  • Incorrect. The domain should be limited to \(x \leq 6\).

The correct option that describes the domain and range of the line segment defined by the points (2, 1) and (6, -3) is:

b. Domain: \(2 < x \le 6\) Range: \(-3 \le y < 1\)