Domain of a function
Problem 13.25 · easy
Find the domain of \( \displaystyle f(x) = \frac{\sqrt{x + 5}}{x - 4} \). Give it in interval notation.
- \[ \left. x + 5 \right|_{\substack{ x=-5 }} = 0 \]At x = -5 the radicand is 0 (it must be ≥ 0).✓ Proved
- \[ \left. x - 4 \right|_{\substack{ x=4 }} = 0 \]At x = 4 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.
Answer \( \left[-5, 4) \cup (4, \infty) \)
Lines: 2 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | 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 (misleading) — The sentence incorrectly references a logarithm, which does not appear in the problem, potentially confusing the reader.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 2026-10-17 — 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: fail (misleading) 2026-10-17 — The sentence incorrectly references a logarithm, which does not appear in the problem, potentially confusing the reader.qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the critical points and states the general conditions for the domain of the function components. The final interval notation correctly reflects these constraints.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the constraints for the square root and the denominator. The final interval notation correctly includes -5 and excludes 4.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.