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Making a piecewise function continuous

Problem 1.297 · medium

Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k x + 4 & x < 3 \\ x^{2} - 2 x - 1 & x \ge 3 \end{cases} \) continuous for all \( \displaystyle x \).

Each piece is a polynomial, so f is continuous everywhere except possibly at x = 3. There the left-hand limit, the right-hand limit and f(c) must agree.
Reviewed
\[ \lim_{x \to 3^-}\left(k x + 4\right) = 3 k + 4 \]
The left-hand limit, in terms of k.✓ Proved
\[ \lim_{x \to 3^+}\left(x^{2} - 2 x - 1\right) = 2 \]
The right-hand limit, which is also f(c).✓ Proved
\[ 2 \]
Setting them equal gives k = -2/3.✓ Proved

Answer \( k = - \frac{2}{3} \)

Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see the receipt.

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	claude-sonnet-5-5	claude-sonnet-5-5: This is a sentence, not an equation. Both pieces are polynomials, so f is continuous on x < 3 and on x > 3. Continuity at x = 3 requires the left limit, the right limit and f(3) to agree. That is the correct condition.
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	✓ 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	with that k the two pieces agree at c ± 1e-10

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution claims that setting the limits equal results in the equation '2 = 2', which is incorrect. The correct equation is 3k + 4 = 2, which leads to k = -2/3. The stated equation does not involve k and does not logically lead to the answer.

Senior review claude-sonnet-5-5, 2026-10-05: fail — The answer k = -2/3 is correct, since 3(-2/3) + 4 = 2 matches the right-hand limit 9 - 6 - 1 = 2. But the final line is 2 = 2, and the equation 3k + 4 = 2 and its solution never appear, so a student cannot see how k is found.

qwen3.6:27b-mlx: uphold — The displayed line '2 = 2' is a tautology that drops k, so the equation 3k + 4 = 2 that actually determines k = -2/3 is never written and the solve step is skipped.

Every verdict on record (5)

qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution claims that setting the limits equal results in the equation '2 = 2', which is incorrect. The correct equation is 3k + 4 = 2, which leads to k = -2/3. The stated equation does not involve k and does not logically lead to the answer.
gpt-oss:20b: pass 2026-10-05
claude-sonnet-5-5: fail (misleading) 2026-10-05 — The answer k = -2/3 is correct, since 3(-2/3) + 4 = 2 matches the right-hand limit 9 - 6 - 1 = 2. But the final line is 2 = 2, and the equation 3k + 4 = 2 and its solution never appear, so a student cannot see how k is found.
qwen3.6:27b-mlx: fail (error) 2026-10-05 — Step 4 claims that setting the limits equal yields k = -2/3, but the equation shown is '2 = 2', which is a tautology and does not involve k. The correct equation should be 3k + 4 = 2.
gpt-oss:20b: pass 2026-10-05

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/continuity_parameter, checked 2026-10-05 with SymPy 1.14.0.