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Inverse functions

Problem 13.225 · easy

Find the inverse of \( \displaystyle f(x) = 6 - x \).

Write y = f(x) and solve for x. Subtract the constant and divide. Then rename y as x.
\[ \left. 6 - x \right|_{\substack{ x=6 - x }} = x \]
f(f⁻¹(x)) = x, so this is the inverse.✓ Proved
\[ \left. 6 - x \right|_{\substack{ x=\left. 6 - x \right|_{\substack{ x=2 }} }} = 2 \]
And f⁻¹(f(2)) = 2.✓ Proved

Answer \( f^{-1}(x) = 6 - x \)

Lines: 2 proved, 1 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
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	f then the answer, in floating point, returns the starting x

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution claims f(x) = 6 - x is its own inverse, which is false. The correct inverse is f^{-1}(x) = 6 - x only if f(x) = 6 - x is an involution, but f(f(x)) = 6 - (6 - x) = x, so it IS its own inverse. Wait, let me re-evaluate. f(x) = 6 - x. f(f(x)) = 6 - (6 - x) = x. Yes, it is its own inverse. The algebra in step 2 checks out. Step 3 checks out. The sentence in step 1 is vague but the method described (solve for x) is standard. Is there an error? Let's check the inverse calculation. y = 6 - x => x = 6 - y => y = 6 - x. So f^{-1}(x) = 6 - x. The solution is actually correct. Why did I think it was wrong? I confused it with x - 6 or something. f(x) = 6-x is a reflection across x=3. It is an involution. The solution is correct.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution claims f(x) = 6 - x is its own inverse, which is false. The correct inverse is f^{-1}(x) = 6 - x only if f(x) = 6 - x is an involution, but f(f(x)) = 6 - (6 - x) = x, so it IS its own inverse. Wait, let me re-evaluate. f(x) = 6 - x. f(f(x)) = 6 - (6 - x) = x. Yes, it is its own inverse. The algebra in step 2 checks out. Step 3 checks out. The sentence in step 1 is vague but the method described (solve for x) is standard. Is there an error? Let's check the inverse calculation. y = 6 - x => x = 6 - y => y = 6 - x. So f^{-1}(x) = 6 - x. The solution is actually correct. Why did I think it was wrong? I confused it with x - 6 or something. f(x) = 6-x is a reflection across x=3. It is an involution. The solution is correct.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution claims f(x) = 6 - x is its own inverse, which is mathematically false (the inverse is 6 - x only if f(x) = x, but f(f(x)) = x - 6). The verification steps are circular or incorrect: checking f(f(x)) = x is necessary but not sufficient to define the inverse function explicitly, and the specific check f(f(2)) = 2 is false since 6-(6-2) = 2 is true, but f(f(x)) = x implies f is an involution, which 6-x is not (6-(6-x) = x, wait. Let's re-evaluate. f(x) = 6-x. f(f(x)) = 6-(6-x) = x. So f IS its own inverse. My initial thought was wrong. 6-x is an involution. Let me re-read carefully. f(x) = 6-x. Inverse: y = 6-x => x = 6-y => y = 6-x. So f^-1(x) = 6-x. The stated answer is correct. The solution says 'Subs(6-x, x, 6-x) = x'. This means f(f(x)) = x. This is true. The second check is f(f(2)) = 2. This is true. The sentence says 'Write y=f(x) and solve for x... Then rename y as x.' This is the standard method. The result is correct. Why did I think it was wrong? 6-x is indeed its own inverse. Example: f(1)=5, f(5)=1. Correct. So the solution is actually correct. Wait, let me double check the 'Subs' notation. Subs(6-x, x, 6-x) replaces x with 6-x in 6-x, resulting in 6-(6-x) = x. This confirms f(f(x))=x. This proves f is its own inverse. The solution is sound.
gpt-oss:20b: fail (misleading) 2026-10-16 — The first sentence incorrectly describes solving for the inverse: solving y=6−x gives x=6−y, so f⁻¹(x)=6−x, not x=6−x. This misleads a student about the correct algebraic steps.
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution incorrectly claims that f(x) = 6 - x is its own inverse. The actual inverse is f^{-1}(x) = 6 - x, which happens to be the same function, but the reasoning in step 1 is flawed because it suggests a generic procedure ('subtract constant and divide') that does not apply here (there is no division). More importantly, step 2 and 3 are not derivations but checks, and they fail to actually derive the inverse. The prompt asks to 'Find' the inverse, not just verify a guess. The provided 'solution' does not show the derivation steps (y = 6-x => x = 6-y => y = 6-x) but rather asserts the answer and checks it. However, the main error is that the text in step 1 describes a procedure ('divide') that is not used and is irrelevant, making the instructional content misleading. Actually, looking closer, f(x) = 6-x IS its own inverse. f(f(x)) = 6-(6-x) = x. So the answer is correct. But the method described in sentence 1 is 'Subtract the constant and divide'. For y=6-x, you subtract 6 to get y-6=-x, then divide by -1 to get x = (6-y)/-1 = y-6? No, x = 6-y. The sentence says 'divide', implying a coefficient other than 1 or -1 might be involved, but here you just multiply by -1. The bigger issue is that the solution doesn't actually perform the algebraic manipulation to find the inverse; it just states the answer and verifies it. This is not a valid 'solution' to 'Find' the inverse. It's a verification. A student would learn that you can just guess and check, or that the procedure involves division which isn't shown. Let's look at the severity. Is it an error? The final answer is correct. The steps are verification, not derivation. The sentence 1 is misleading because it describes a generic linear inverse process that doesn't match the specific algebraic steps shown (none are shown). It's misleading instruction.
gpt-oss:20b: pass 2026-10-15

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/inverse_function, checked 2026-10-18 with SymPy 1.14.0.