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Composition of functions

Problem 13.4 · medium

Let \( \displaystyle f(x) = 2 x + 1 \) and \( \displaystyle g(x) = x^{2} + 1 \). 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
\[ \left. 2 x + 1 \right|_{\substack{ x=x^{2} + 1 }} = 2 x^{2} + 3 \]
f(g(x)), simplified.✓ Proved
\[ \left. x^{2} + 1 \right|_{\substack{ x=2 x + 1 }} = \left(2 x + 1\right)^{2} + 1 \]
g(f(x)), simplified.✓ Proved
The two orders give different functions: composition is not commutative.
Reviewed

Answer \( (f \circ g)(x) = 2 x^{2} + 3,\quad (g \circ f)(x) = \left(2 x + 1\right)^{2} + 1 \)

Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. 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: pass
qwen3.6:27b-mlx: pass — The solution correctly defines function composition, performs the substitutions accurately, and provides a valid observation about non-commutativity.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly defines function composition, performs the substitutions accurately, and provides a valid observation about non-commutativity.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer response could not be parsed: { "verdict": "pass", "severity": "", "notes": "The solution correctly identifies the definition of function composition, performs the substitutions accurately, and notes the non-commutative property.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly defines function composition, performs the substitutions accurately, and provides a valid observation about non-commutativity.
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.