Composition of functions
Problem 13.6 · easy
Let \( \displaystyle f(x) = \frac{1}{x - 1} \) and \( \displaystyle g(x) = 4 x - 4 \). Find \( \displaystyle (f \circ g)(0) \).
- (f∘g)(0) = f(g(0)): evaluate g first, then f at the result.Reviewed
- \[ \left. 4 x - 4 \right|_{\substack{ x=0 }} = -4 \]g(0).✓ Proved
- \[ \left. 1 \frac{1}{x - 1} \right|_{\substack{ x=-4 }} = - \frac{1}{5} \]f at that value.✓ Proved
Answer \( (f \circ g)(0) = - \frac{1}{5} \)
✓ 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | g then f evaluated in floating point |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the definition of function composition and performs the arithmetic accurately.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the definition of function composition and performs the arithmetic accurately.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/function_composition, checked 2026-10-17 with SymPy 1.14.0.