Domain of a function
Problem 13.151 · easy
Find the domain of \( \displaystyle f(x) = \sqrt{2 x - 6} \). Give it in interval notation.
- \[ \left. 2 x - 6 \right|_{\substack{ x=3 }} = 0 \]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.
Answer \( \left[3, \infty) \)
Lines: 1 proved, 1 not checked. 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 | 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 (style) — [domain objection, downgraded to style] The second sentence is a generic rule that does not actually identify the domain of the given function; it neither applies the specific condition 2x‑6 ≥ 0 nor concludes the interval [3,∞).qwen3.6:27b-mlx: pass — The solution correctly identifies the condition for the square root (radicand ≥ 0) and solves for the boundary. The stated answer [3, ∞) is correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the condition for the square root (radicand ≥ 0) and solves for the boundary. The stated answer [3, ∞) is correct.gpt-oss:20b: fail (style) 2026-10-18 — [domain objection, downgraded to style] The second sentence is a generic rule that does not actually identify the domain of the given function; it neither applies the specific condition 2x‑6 ≥ 0 nor concludes the interval [3,∞).qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the condition for the square root (radicand >= 0) and solves for the boundary. The general statement in line 2 is accurate and covers the necessary constraints for the function type.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the constraint for the square root function and solves for the boundary. The general rule stated in line 2 is accurate and applicable.gpt-oss:20b: pass 2026-10-14
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.