Problem Statement
Consider the rational 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 the graph of $y = f(x)$.
**(b)** [3 marks] Find an expression for $f^{-1}(x)$, stating its domain clearly.
**(c)** [1 marks] State the coordinates of the $y$-intercept of $y = f(x)$.
Verified Solution & Marking Scheme
Part (a) [A1]
Vertical asymptote: $x = -2$
Part (a) [A1]
Horizontal asymptote: $y = \lim_{x \to \infty} \frac{3x - 1}{x + 2} = 3 \implies y = 3$
Part (b) [M1]
Set $y = \frac{3x - 1}{x + 2}$ and swap variables: $x = \frac{3y - 1}{y + 2}$
Part (b) [M1]
Solve for $y$: $x(y + 2) = 3y - 1 \implies xy + 2x = 3y - 1 \implies y(x - 3) = -2x - 1$
Part (b) [A1]
$f^{-1}(x) = \frac{2x + 1}{3 - x} = \frac{-2x - 1}{x - 3}$ with domain $x \in \mathbb{R}, x \neq 3$
Part (c) [A1]
$f(0) = \frac{3(0) - 1}{0 + 2} = -\frac{1}{2}$, coordinates are $\left(0, -\frac{1}{2}\right)$