Domain of a function
Problem 13.365 · easy
Find the domain of \( \displaystyle f(x) = \frac{\sqrt{x - 2}}{x - 4} \). Give it in interval notation.
- \[ \left. x - 2 \right|_{\substack{ x=2 }} = 0 \]At x = 2 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 \( \left[2, 4) \cup (4, \infty) \)
Lines: 2 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 | ✓ 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 (style) — [domain objection, downgraded to style] The final sentence is a generic rule that does not actually apply the conditions to determine the domain; it fails to state the resulting interval and is therefore incomplete.qwen3.6:27b-mlx: pass — The solution correctly identifies the constraints for the square root and the denominator, and the final interval notation is correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the constraints for the square root and the denominator, and the final interval notation is correct.gpt-oss:20b: fail (style) 2026-10-19 — [domain objection, downgraded to style] The final sentence is a generic rule that does not actually apply the conditions to determine the domain; it fails to state the resulting interval and is therefore incomplete.qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the critical points and states the general conditions for the domain of the given function. The final answer matches the derived constraints.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the critical points and states the general conditions for the domain of the given function. The logic is sound and leads to the correct interval notation.gpt-oss:20b: pass 2026-10-17
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 2026-10-19 with SymPy 1.14.0.