Find the inverse function of f(x)=(3x-5)/(2x+1) and state its domain.

f^-1(y)=(5+2y)/(3-y), domain {y∈R| y≠ 3/2}

Find the inverse function of $f (x) = \frac{3 x - 5}{2 x + 1}$ and state its domain.

medium 2 marks Class 12 · Mathematics · Relations and Functions

Answer

f^{- 1} (y) = \frac{5 + 2 y}{3 - y}

, domain

{y \in R ∣ y \neq = \frac{3}{2}}

The correct answer is $f^{- 1} (y) = \frac{5 + 2 y}{3 - y}$ , domain ${y \in R ∣ y \neq = \frac{3}{2}}$ — 10. Therefore the domain of the inverse is all real numbers except $\frac{3}{2}$ .

Step-by-step solution

1. Set $y = \frac{3 x - 5}{2 x + 1}$ to find $x$ in terms of $y$ .
2. Multiply both sides by $(2 x + 1)$ : $y (2 x + 1) = 3 x - 5$ .
3. Expand: $2 y x + y = 3 x - 5$ .
4. Collect $x$ terms: $2 y x - 3 x = y - 5$ .
5. Factor $x$ : $x (2 y - 3) = y - 5$ .
6. Solve for $x$ : $x = \frac{y - 5}{2 y - 3}$ .
7. Replace $x$ with $f^{- 1} (y)$ : $f^{- 1} (y) = \frac{y - 5}{2 y - 3}$ .
8. Simplify by multiplying numerator and denominator by $- 1$ : $f^{- 1} (y) = \frac{5 + 2 y}{3 - y}$ .
9. The denominator $2 y - 3$ cannot be zero, so $y \neq = \frac{3}{2}$ .
10. Therefore the domain of the inverse is all real numbers except $\frac{3}{2}$ .

Common mistake: Forgetting to exclude the value that makes the denominator zero

Concept tested: Finding inverse of a rational function

Published 6 September 2026 · How this solution was prepared and checked

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Find the inverse function of $f (x) = \frac{3 x - 5}{2 x + 1}$ and state its domain.