Domain of a function
Problem 13.369 · easy
Find the domain of \( \displaystyle f(x) = \sqrt{- 2 x - 4} \). Give it in interval notation.
- \[ \left. - 2 x - 4 \right|_{\substack{ x=-2 }} = 0 \]At x = -2 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.
Answer \( (-\infty, -2\right] \)
Lines: 1 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Not checked | — | 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: fail (error) — The solution omits the key inequality -2x-4≥0 and its solution x≤-2, and the generic sentence about denominators, roots, and logs is irrelevant to this problem. Thus the setup and conclusion are incomplete and incorrect.qwen3.6:27b-mlx: fail (misleading) — The sentence lists conditions for fractions and logarithms which are irrelevant to this problem, potentially confusing the student about what constraints apply to square roots.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-10-19 — The sentence lists conditions for fractions and logarithms which are irrelevant to this problem, potentially confusing the student about what constraints apply to square roots.gpt-oss:20b: fail (error) 2026-10-19 — The solution omits the key inequality -2x-4≥0 and its solution x≤-2, and the generic sentence about denominators, roots, and logs is irrelevant to this problem. Thus the setup and conclusion are incomplete and incorrect.qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the boundary point and states the general principle for determining the domain of a square root function. The logic is sound and leads to the correct interval.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (misleading) 2026-10-17 — Sentence 2 lists conditions for fractions and logarithms which are irrelevant to this problem, potentially confusing the student about the necessary steps. It also fails to explicitly state the inequality -2x - 4 >= 0 that must be solved.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.