Domain of a function
Problem 13.362 · easy
Find the domain of \( \displaystyle f(x) = \sqrt{- x - 3} \). Give it in interval notation.
- \[ \left. - x - 3 \right|_{\substack{ x=-3 }} = 0 \]At x = -3 the radicand is 0 (it must be ≥ 0).✓ Proved
- Keep the x where every fraction has a nonzero denominator, every square root a nonnegative radicand and every logarithm a positive argument.Reviewed
Answer \( (-\infty, -3\right] \)
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | f evaluated at 321 points in [−10, 10] is real exactly where the claimed domain says |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the boundary point and states the general condition for the domain of a square root function. The logic is sound and leads to the correct answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the boundary point and states the general condition for the domain of a square root function. The logic is sound and leads to the correct answer.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the boundary point and states the general condition for the domain of a square root function. The final answer matches the derived condition.gpt-oss:20b: fail (style) 2026-10-17 — [domain objection, downgraded to style] The second sentence is a generic statement about fractions, square roots, and logarithms that does not address the specific problem of finding the domain of √{-x-3}. It neither justifies the domain nor follows from the preceding line, so it is unnecessary and could confuse a student. The solution should explicitly state that the radicand must be non‑negative, leading to x≤-3, and then give the domain as (-∞, -3].qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the boundary point and states the general condition for the domain of a square root function. The final answer matches the derived interval.gpt-oss:20b: pass 2026-10-17
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/domain_range, checked 2026-10-19 with SymPy 1.14.0.