Home›Calculus 1›Domain of a function›Problem 13.22

Domain of a function

Problem 13.22 · 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 \( (-\infty, -4\right] \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the boundary point and states the general condition for the domain of a square root function. Although the problem does not involve fractions or logarithms, the general statement is not incorrect, just broader than necessary.

Every verdict on record (6)

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. Although the problem does not involve fractions or logarithms, the general statement is not incorrect, just broader than necessary.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-14 — 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 2026-10-14
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the condition for the square root (radicand >= 0) and solves for the boundary. The general sentence in step 2 is accurate, although it mentions fractions and logarithms which are not present in this specific problem, it does not lead to an incorrect conclusion.
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 2026-10-17 with SymPy 1.14.0.