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Domain of a function

Problem 13.209 · easy

Find the domain of \( \displaystyle f(x) = \frac{\sqrt{x + 3}}{x - 5} \). Give it in interval notation.

\[ \left. x + 3 \right|_{\substack{ x=-3 }} = 0 \]
At x = -3 the radicand is 0 (it must be ≥ 0).✓ Proved
\[ \left. x - 5 \right|_{\substack{ x=5 }} = 0 \]
At x = 5 the denominator is 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 \( \left[-3, 5) \cup (5, \infty) \)

Lines: 2 proved, 1 reviewed. 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	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	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 critical points and applies the standard domain constraints for square roots and rational functions. The resulting interval notation is correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the critical points and applies the standard domain constraints for square roots and rational functions. The resulting interval notation is correct.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the constraints for the square root and the denominator, and the final interval notation accurately reflects these conditions.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the constraints for the square root and the denominator, and the stated answer correctly combines these into the proper interval notation.
gpt-oss:20b: inconclusive 2026-10-15 — reviewer response could not be parsed: {"verdict":"fail","severity":"style","notes":"The final sentence is overly generic and does not actually state the domain; it also mentions logarithms which are irrelevant to the problem. The solution

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.