Domain of a function
Problem 13.207 · easy
Find the domain of \( \displaystyle f(x) = \sqrt{3 x + 9} \). Give it in interval notation.
- \[ \left. 3 x + 9 \right|_{\substack{ x=-3 }} = 0 \]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 \( \left[-3, \infty) \)
✓ 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 domain constraint for the square root function and calculates the boundary point. The general statement in line 2 is accurate, although it mentions fractions and logarithms which are not present in this specific problem; this is harmless general advice rather than an error.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the domain constraint for the square root function and calculates the boundary point. The general statement in line 2 is accurate, although it mentions fractions and logarithms which are not present in this specific problem; this is harmless general advice rather than an error.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the condition for the square root and solves for the boundary. The general statement in line 2 is accurate, though only the square root condition is relevant here.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the condition for the square root (radicand ≥ 0) and solves for the boundary. The stated answer is correct.gpt-oss:20b: pass 2026-10-15
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 2026-10-18 with SymPy 1.14.0.