Inverse functions
Problem 13.456 · easy
Find the inverse of \( \displaystyle f(x) = - 5 x - 1 \).
- Write y = f(x) and solve for x. Subtract the constant and divide. Then rename y as x.
- \[ \left. - 5 x - 1 \right|_{\substack{ x=\frac{\left(-1\right) x}{5} - 1 \cdot \frac{1}{5} }} = x \]f(f⁻¹(x)) = x, so this is the inverse.✓ Proved
- \[ \left. \frac{\left(-1\right) x}{5} - 1 \cdot \frac{1}{5} \right|_{\substack{ x=\left. - 5 x - 1 \right|_{\substack{ x=1 }} }} = 1 \]And f⁻¹(f(1)) = 1.✓ Proved
Answer \( f^{-1}(x) = - \frac{x}{5} - \frac{1}{5} \)
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: passqwen3.6:27b-mlx: fail (misleading) — The solution claims to 'rename y as x' but presents the final answer as f^{-1}(x) = -x/5 - 1/5, which is the correct inverse function. However, the sentence 'Subtract the constant and divide' is ambiguous and potentially misleading regarding the order of operations (one must add 1 then divide by -5, not just 'subtract the constant'). More critically, the verification steps are presented as equations that 'prove' the inverse, but they only check specific points or identities without deriving the general form, making the logical flow from step 1 to the stated answer opaque. The primary issue is that step 1 describes a process but doesn't show the algebraic manipulation, and the description 'Subtract the constant' is technically incorrect for the operation y = -5x - 1 (you add 1 to isolate the term with x).
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-10-30 — The solution claims to 'rename y as x' but presents the final answer as f^{-1}(x) = -x/5 - 1/5, which is the correct inverse function. However, the sentence 'Subtract the constant and divide' is ambiguous and potentially misleading regarding the order of operations (one must add 1 then divide by -5, not just 'subtract the constant'). More critically, the verification steps are presented as equations that 'prove' the inverse, but they only check specific points or identities without deriving the general form, making the logical flow from step 1 to the stated answer opaque. The primary issue is that step 1 describes a process but doesn't show the algebraic manipulation, and the description 'Subtract the constant' is technically incorrect for the operation y = -5x - 1 (you add 1 to isolate the term with x).gpt-oss:20b: pass 2026-10-30qwen3.6:27b-mlx: pass 2026-10-29 — The solution correctly applies the standard algebraic method for finding the inverse of a linear function and verifies the result with composition checks.gpt-oss:20b: pass 2026-10-29qwen3.6:27b-mlx: pass 2026-10-29 — The solution correctly applies the standard method for finding the inverse of a linear function. The verification steps confirm the result is correct.gpt-oss:20b: pass 2026-10-29
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 2026-10-30 with SymPy 1.14.0.