Domain of a function
Problem 13.149 · easy
Find the domain of \( \displaystyle f(x) = \frac{x + 3}{x^{2} - 16} \). Give it in interval notation.
- \[ \left. x^{2} - 16 \right|_{\substack{ x=-4 }} = 0 \]At x = -4 the denominator is 0.✓ Proved
- \[ \left. x^{2} - 16 \right|_{\substack{ x=4 }} = 0 \]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.Reviewed
Answer \( (-\infty, -4) \cup (-4, 4) \cup (4, \infty) \)
✓ Nihil obstat Lines: 2 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 | ✓ 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: passqwen3.6:27b-mlx: pass — The solution correctly identifies the points where the denominator is zero and states the general principle for finding the domain of rational functions. The final answer is correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the points where the denominator is zero and states the general principle for finding the domain of rational functions. The final answer is correct.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the points where the denominator is zero and states the general principle for determining the domain of rational functions. The final answer is correct.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the points where the denominator is zero and states the general principle for determining the domain of rational functions. The final answer is correct.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 2026-10-18 with SymPy 1.14.0.