Composition of functions
Problem 13.157 · medium
Let \( \displaystyle f(x) = - 2 x^{2} - 4 x \) and \( \displaystyle g(x) = x^{2} + 3 \). 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
- \[ \left. - 2 x^{2} - 4 x \right|_{\substack{ x=x^{2} + 3 }} = - 2 x^{4} - 16 x^{2} - 30 \]f(g(x)), simplified.✓ Proved
- \[ \left. x^{2} + 3 \right|_{\substack{ x=- 2 x^{2} - 4 x }} = 4 x^{2} \left(x + 2\right)^{2} + 3 \]g(f(x)), simplified.✓ Proved
- The two orders give different functions: composition is not commutative.Reviewed
Answer \( (f \circ g)(x) = - 2 x^{4} - 16 x^{2} - 30,\quad (g \circ f)(x) = 4 x^{2} \left(x + 2\right)^{2} + 3 \)
✓ 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 simplifies the results correctly. The concluding remark is mathematically sound.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly defines function composition, performs the substitutions accurately (verified by the algebra check), and simplifies the results correctly. The concluding remark is mathematically sound.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly defines function composition, performs the substitutions accurately, and simplifies the results as requested. The final observation about non-commutativity is correct and relevant.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly defines function composition and provides the correct simplified results for both (f ∘ g)(x) and (g ∘ f)(x). The concluding remark 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.