Inverse functions
Problem 13.455 · medium
Find the inverse of \( \displaystyle f(x) = \frac{2 x + 5}{3 x - 1} \).
- Write y = f(x) and solve for x. Clear the denominator, collect the x terms, and factor x out. Then rename y as x.
- \[ \left. \frac{2 x + 5}{3 x - 1} \right|_{\substack{ x=\frac{x + 5}{3 x - 2} }} = x \]f(f⁻¹(x)) = x, so this is the inverse.✓ Proved
- \[ \left. \frac{x + 5}{3 x - 2} \right|_{\substack{ x=\left. \frac{2 x + 5}{3 x - 1} \right|_{\substack{ x=2 }} }} = 2 \]And f⁻¹(f(2)) = 2.✓ Proved
Answer \( f^{-1}(x) = \frac{x + 5}{3 x - 2} \)
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 (error) — The stated answer is incorrect; the correct inverse is (x + 5)/(3x - 2) only if the original function was different, but for f(x) = (2x+5)/(3x-1), the inverse is (x+5)/(3x-2) is actually WRONG. Let's re-calculate: y = (2x+5)/(3x-1) => y(3x-1) = 2x+5 => 3xy - y = 2x + 5 => 3xy - 2x = y + 5 => x(3y - 2) = y + 5 => x = (y+5)/(3y-2). So f^-1(x) = (x+5)/(3x-2). Wait, the stated answer IS correct. Why did I think it was wrong? Let me re-read the prompt. Problem: f(x) = (2x+5)/(3x-1). Inverse: swap x and y? No, solve for x. x = (y+5)/(3y-2). So f^-1(x) = (x+5)/(3x-2). The stated answer is correct. The solution steps are: 1. Sentence describing the method. 2. Equation checking f(f^-1(x)) = x. 3. Equation checking f^-1(f(2)) = 2. The sentences are just descriptions. Is there an error? The sentence says 'rename y as x'. This is standard. The equations are marked 'proved'. The verdict should be pass? Let me double check the inverse calculation again. y = (2x+5)/(3x-1). 3xy - y = 2x + 5. 3xy - 2x = y + 5. x(3y - 2) = y + 5. x = (y+5)/(3y-2). f^-1(x) = (x+5)/(3x-2). The stated answer is correct. The solution is sound. Why would it fail? Maybe the sentence is too vague? 'Clear the denominator, collect the x terms, and factor x out. Then rename y as x.' This is a valid description of the algebraic steps. The equations verify the result. It seems correct. Wait, look at the equation 2: Subs((2*x + 5)/(3*x - 1), x, (x + 5)/(3*x - 2)) = x. This checks f(f^-1(x)) = x. Equation 3 checks f^-1(f(2)) = 2. This is sufficient. The solution is correct. Verdict: pass.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-30 — The stated answer is incorrect; the correct inverse is (x + 5)/(3x - 2) only if the original function was different, but for f(x) = (2x+5)/(3x-1), the inverse is (x+5)/(3x-2) is actually WRONG. Let's re-calculate: y = (2x+5)/(3x-1) => y(3x-1) = 2x+5 => 3xy - y = 2x + 5 => 3xy - 2x = y + 5 => x(3y - 2) = y + 5 => x = (y+5)/(3y-2). So f^-1(x) = (x+5)/(3x-2). Wait, the stated answer IS correct. Why did I think it was wrong? Let me re-read the prompt. Problem: f(x) = (2x+5)/(3x-1). Inverse: swap x and y? No, solve for x. x = (y+5)/(3y-2). So f^-1(x) = (x+5)/(3x-2). The stated answer is correct. The solution steps are: 1. Sentence describing the method. 2. Equation checking f(f^-1(x)) = x. 3. Equation checking f^-1(f(2)) = 2. The sentences are just descriptions. Is there an error? The sentence says 'rename y as x'. This is standard. The equations are marked 'proved'. The verdict should be pass? Let me double check the inverse calculation again. y = (2x+5)/(3x-1). 3xy - y = 2x + 5. 3xy - 2x = y + 5. x(3y - 2) = y + 5. x = (y+5)/(3y-2). f^-1(x) = (x+5)/(3x-2). The stated answer is correct. The solution is sound. Why would it fail? Maybe the sentence is too vague? 'Clear the denominator, collect the x terms, and factor x out. Then rename y as x.' This is a valid description of the algebraic steps. The equations verify the result. It seems correct. Wait, look at the equation 2: Subs((2*x + 5)/(3*x - 1), x, (x + 5)/(3*x - 2)) = x. This checks f(f^-1(x)) = x. Equation 3 checks f^-1(f(2)) = 2. This is sufficient. The solution is correct. Verdict: pass.gpt-oss:20b: pass 2026-10-30qwen3.6:27b-mlx: pass 2026-10-29 — The solution correctly identifies the method for finding the inverse and provides valid verification steps. The stated answer is correct.gpt-oss:20b: pass 2026-10-29qwen3.6:27b-mlx: pass 2026-10-29 — The solution correctly identifies the method for finding the inverse and verifies the result using composition. Although the intermediate algebraic steps are omitted, the final answer is correct and the verification confirms it.gpt-oss:20b: fail (misleading) 2026-10-29 — The solution never actually solves for x; it merely states a substitution check without showing the algebraic steps, so a student would be misled into thinking the inverse is verified when it is not demonstrated.
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.