Composition of functions
Problem 13.433 · medium
Let \( \displaystyle f(x) = x^{2} + 3 \) and \( \displaystyle g(x) = - 2 x^{2} - 2 x \). 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
- \[ \left. x^{2} + 3 \right|_{\substack{ x=- 2 x^{2} - 2 x }} = 4 x^{2} \left(x + 1\right)^{2} + 3 \]f(g(x)), simplified.✓ Proved
- \[ \left. - 2 x^{2} - 2 x \right|_{\substack{ x=x^{2} + 3 }} = - 2 x^{4} - 14 x^{2} - 24 \]g(f(x)), simplified.✓ Proved
- The two orders give different functions: composition is not commutative.Reviewed
Answer \( (f \circ g)(x) = 4 x^{2} \left(x + 1\right)^{2} + 3,\quad (g \circ f)(x) = - 2 x^{4} - 14 x^{2} - 24 \)
✓ 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 defines function composition, performs the substitutions accurately (verified by the algebra check), and provides a correct concluding remark about non-commutativity.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-22 — The solution correctly defines function composition, performs the substitutions accurately (verified by the algebra check), and provides a correct concluding remark about non-commutativity.gpt-oss:20b: pass 2026-10-22qwen3.6:27b-mlx: pass 2026-10-21 — The solution correctly defines function composition, performs the substitutions accurately as verified by the algebra system, and provides a correct concluding remark about non-commutativity.gpt-oss:20b: pass 2026-10-21qwen3.6:27b-mlx: pass 2026-10-21 — The solution correctly defines function composition, performs the algebraic substitutions accurately, and provides a valid observation about non-commutativity.gpt-oss:20b: pass 2026-10-21
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 2026-10-22 with SymPy 1.14.0.