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

f^-1(x)=(3x-2)/(x+5), with domain x≠ -5

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

hard 5 marks Class 12 · Mathematics · Relations and Functions

Answer

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

, with domain

x \neq = - 5

The correct answer is $f^{- 1} (x) = \frac{3 x - 2}{x + 5}$ , with domain $x \neq = - 5$ — State the final answer with domain restriction.

Write $y = \frac{5 x + 2}{3 - x}$ and aim to solve for $x$ in terms of $y$ .
Cross‑multiply: $y (3 - x) = 5 x + 2$ .
Distribute $y$ : $3 y - y x = 5 x + 2$ .
Collect terms containing $x$ on one side: $- y x - 5 x = 2 - 3 y$ .
Factor $x$ out: $x (- y - 5) = 2 - 3 y$ .
Solve for $x$ : $x = \frac{2 - 3 y}{- y - 5} = \frac{3 y - 2}{y + 5}$ (multiply numerator and denominator by $- 1$ ).
Replace $y$ by $x$ to obtain the inverse function: $f^{- 1} (x) = \frac{3 x - 2}{x + 5}$ .
Determine the domain of $f^{- 1}$ : the denominator $x + 5 \neq = 0$ , so $x \neq = - 5$ .
State the final answer with domain restriction.

Common mistake: Failing to multiply numerator and denominator by

- 1

after isolating

x

, leading to a sign error in the inverse.

Concept tested: Finding inverse of rational function