Composition of functions
Problem 13.160 · medium
Let \( \displaystyle f(x) = \frac{1}{x - 2} \) and \( \displaystyle g(x) = - 3 x^{2} - 3 x \). Find \( \displaystyle (f \circ g)(x) \) and \( \displaystyle (g \circ f)(x) \), simplified.
- (f∘g)(x) = f(g(x)): put g(x) in place of x in f. (g∘f) is the other way round.Reviewed
- \[ \left. 1 \frac{1}{x - 2} \right|_{\substack{ x=- 3 x^{2} - 3 x }} = - \frac{1}{3 x^{2} + 3 x + 2} \]f(g(x)), simplified.✓ Proved
- \[ \left. - 3 x^{2} - 3 x \right|_{\substack{ x=1 \frac{1}{x - 2} }} = \frac{3 - 3 x}{x^{2} - 4 x + 4} \]g(f(x)), simplified.✓ Proved
- The two orders give different functions: composition is not commutative.Reviewed
Answer \( (f \circ g)(x) = - \frac{1}{3 x^{2} + 3 x + 2},\quad (g \circ f)(x) = \frac{3 \left(1 - x\right)}{x^{2} - 4 x + 4} \)
✓ Nihil obstat Lines: 2 proved, 2 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 |
| 4 | 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(g(x)) and g(f(x)) computed in floating point at sample points |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies the definition of function composition and the algebraic simplifications are verified. The concluding remark is accurate.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies the definition of function composition and the algebraic simplifications are verified. The concluding remark is accurate.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the composition operations, the algebraic substitutions are verified as correct, and the concluding remark is accurate.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the definition of function composition and provides the correct simplified results for both (f ∘ g)(x) and (g ∘ f)(x). The final observation about non-commutativity is accurate.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 2026-10-18 with SymPy 1.14.0.