Problem Statement
Consider the function $f(x) = \frac{3x + 1}{x - 2}$ for $x \in \mathbb{R}, x \neq 2$.
**(a)** [2 marks] Write down the equations of the vertical and horizontal asymptotes of $y = f(x)$.
**(b)** [4 marks] Find an expression for $f^{-1}(x)$, and state its domain and range.
Verified Solution & Marking Scheme
Part (a) [A1]
Vertical asymptote occurs at denominator zero: $x = 2$
Part (a) [A1]
Horizontal asymptote is $\lim_{x \to \pm \infty} \frac{3x+1}{x-2} = 3 \implies y = 3$
Part (b) [M1]
Set $x = \frac{3y + 1}{y - 2}$ and rearrange: $x(y - 2) = 3y + 1$
Part (b) [M1]
$xy - 2x = 3y + 1 \implies xy - 3y = 2x + 1 \implies y(x - 3) = 2x + 1$
Part (b) [A1]
$f^{-1}(x) = \frac{2x + 1}{x - 3}$
Part (b) [A1]
Domain: $\{x \in \mathbb{R} \mid x \neq 3\}$; Range: $\{y \in \mathbb{R} \mid y \neq 2\}$