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Composition of functions

Problem 13.3 · easy

Let \( \displaystyle f(x) = 4 x + 1 \) and \( \displaystyle g(x) = \sqrt{x + 5} \). Find \( \displaystyle (f \circ g)(3) \).

(f∘g)(3) = f(g(3)): evaluate g first, then f at the result.
Reviewed
\[ \left. \sqrt{x + 5} \right|_{\substack{ x=3 }} = 2 \sqrt{2} \]
g(3).✓ Proved
\[ \left. 4 x + 1 \right|_{\substack{ x=2 \sqrt{2} }} = 1 + 8 \sqrt{2} \]
f at that value.✓ Proved

Answer \( (f \circ g)(3) = 1 + 8 \sqrt{2} \)

✓ Nihil obstat Lines: 2 proved, 1 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
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	g then f evaluated in floating point

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the definition of function composition and the algebraic steps are verified.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies the definition of function composition and the algebraic steps are verified.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the definition of function composition and the algebraic steps are verified.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the definition of function composition and performs the arithmetic accurately.
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.