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Composition of functions

Problem 13.352 · medium

Let \( \displaystyle f(x) = \sqrt{x + 4} \) and \( \displaystyle g(x) = x^{3} - 4 \). 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.
\[ \left. \sqrt{x + 4} \right|_{\substack{ x=x^{3} - 4 }} = \sqrt{x^{3}} \]
f(g(x)), simplified.✓ Proved
\[ \left. x^{3} - 4 \right|_{\substack{ x=\sqrt{x + 4} }} = \left(x + 4\right)^{\frac{3}{2}} - 4 \]
g(f(x)), simplified.✓ Proved
The two orders give different functions: composition is not commutative.

Answer \( (f \circ g)(x) = \sqrt{x^{3}},\quad (g \circ f)(x) = \left(x + 4\right)^{\frac{3}{2}} - 4 \)

Lines: 2 proved, 2 not checked. 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	Not checked	—	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	Not checked	—	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: fail (error) — The simplification in line 2 is incorrect; f(g(x)) is sqrt(x^3 - 4), not sqrt(x^3). The CAS output shown is mathematically wrong for the given functions.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-19 — The simplification in line 2 is incorrect; f(g(x)) is sqrt(x^3 - 4), not sqrt(x^3). The CAS output shown is mathematically wrong for the given functions.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The simplification in line 2 is incorrect; f(g(x)) simplifies to sqrt(x^3 - 4), not sqrt(x^3). The CAS output shown in line 2 appears to be a hallucination or error, as substituting x^3 - 4 into sqrt(x+4) does not yield sqrt(x^3).
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The simplification in step 2 is incorrect; f(g(x)) should be sqrt(x^3 - 4), not sqrt(x^3). The solution erroneously dropped the '-4' inside the square root.
gpt-oss:20b: pass 2026-10-16

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 2026-10-19 with SymPy 1.14.0.