Question
9) Give the domain and range of the relation \( y=-1+\sqrt{x-2} \)
Ask by Chadwick Mitchell. in Qatar
Feb 24,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
Domain: [2, ∞)
Range: [-1, ∞)
Solution
To find the domain and range of the relation y = -1 + √(x - 2), follow these steps:
1. Domain:
- The square root √(x - 2) is defined only when the expression under the square root is non-negative.
- Set the inside of the square root ≥ 0: x - 2 ≥ 0.
- Solve for x: x ≥ 2.
- Thus, the domain is [2, ∞).
2. Range:
- The smallest value for y occurs when x takes its minimum value in the domain.
- When x = 2, √(2 - 2) = √0 = 0, so y = -1 + 0 = -1.
- As x increases beyond 2, √(x - 2) becomes positive and increases without bound, so y also increases without bound.
- Thus, the range is [-1, ∞).
Final Answer:
Domain: [2, ∞)
Range: [-1, ∞)
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Simplify this solution Bonus Knowledge
To determine the domain of the relation \( y = -1 + \sqrt{x - 2} \), we need the expression under the square root to be non-negative, which means \( x - 2 \geq 0 \). This implies that \( x \geq 2 \), so the domain is \( [2, \infty) \). For the range, when \( x = 2 \), \( y = -1 + \sqrt{2 - 2} = -1 + 0 = -1 \). As \( x \) increases, \( \sqrt{x - 2} \) also increases, meaning \( y \) goes from -1 to infinity. Therefore, the range is \( [-1, \infty) \).
