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Composition of functions

Problem 13.154 · medium

Let \( \displaystyle f(x) = x^{2} - 5 \) and \( \displaystyle g(x) = \frac{1}{x + 1} \). 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
\[ \left. x^{2} - 5 \right|_{\substack{ x=1 \frac{1}{x + 1} }} = -5 + \frac{1}{\left(x + 1\right)^{2}} \]
f(g(x)), simplified.✓ Proved
\[ \left. 1 \frac{1}{x + 1} \right|_{\substack{ x=x^{2} - 5 }} = \frac{1}{x^{2} - 4} \]
g(f(x)), simplified.✓ Proved
The two orders give different functions: composition is not commutative.
Reviewed

Answer \( (f \circ g)(x) = -5 + \frac{1}{\left(x + 1\right)^{2}},\quad (g \circ f)(x) = \frac{1}{x^{2} - 4} \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly computes both compositions and simplifies them. The final sentence is a valid observation about non-commutativity.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly computes both compositions and simplifies them. The final sentence is a valid observation about non-commutativity.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly computes the composite functions and simplifies them. The final sentence is a valid observation about function composition.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly computes both compositions and simplifies them. The final sentence is a correct observation about function composition.
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.