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Domain of a function

Problem 13.440 · easy

Find the domain of \( \displaystyle f(x) = \sqrt{2 x - 8} \). Give it in interval notation.

\[ \left. 2 x - 8 \right|_{\substack{ x=4 }} = 0 \]
At x = 4 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 \( \left[4, \infty) \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the constraint for the square root function and solves for the boundary. The general statement in step 2 is accurate and covers the necessary conditions for domain determination.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-27 — The solution correctly identifies the constraint for the square root function and solves for the boundary. The general statement in step 2 is accurate and covers the necessary conditions for domain determination.
gpt-oss:20b: pass 2026-10-27
qwen3.6:27b-mlx: pass 2026-10-26 — The solution correctly identifies the constraint for the square root function and calculates the boundary point. The general rule stated in sentence 2 is accurate and applicable.
gpt-oss:20b: fail (style) 2026-10-26 — [domain objection, downgraded to style] The solution does not actually solve the inequality 2x‑8 ≥ 0 to obtain x ≥ 4; it only states a generic rule about radicands. The domain conclusion is missing.
qwen3.6:27b-mlx: pass 2026-10-26 — 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, even though denominators and logarithms are not present in this specific problem.
gpt-oss:20b: pass 2026-10-26

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 2026-10-27 with SymPy 1.14.0.