Problem Statement
Let $f(x) = \frac{3x - 5}{x - 2}$ for $x \in \mathbb{R}, x \neq 2$.
(a) Write down the equations of the vertical and horizontal asymptotes. [2 marks]
(b) Find an expression for $f^{-1}(x)$ and state its domain and range. [4 marks]
Verified Solution & Marking Scheme
Part (a): Identify asymptotes
Vertical asymptote occurs where the denominator is zero: $x - 2 = 0 \implies x = 2$.
Horizontal asymptote as $x \to \pm\infty$: $\lim_{x \to \infty} \frac{3x - 5}{x - 2} = \lim_{x \to \infty} \frac{3 - 5/x}{1 - 2/x} = 3 \implies y = 3$.
Part (b): Find inverse function f⁻¹(x)
Let $y = \frac{3x - 5}{x - 2}$. Swap $x$ and $y$:
$$x = \frac{3y - 5}{y - 2} \implies x(y - 2) = 3y - 5$$
$$xy - 2x = 3y - 5 \implies xy - 3y = 2x - 5$$
$$y(x - 3) = 2x - 5 \implies y = \frac{2x - 5}{x - 3}$$
Therefore, $f^{-1}(x) = \frac{2x - 5}{x - 3}$.
Domain and Range of f⁻¹
Domain of $f^{-1}$ is the range of $f$: $x \in \mathbb{R}, x \neq 3$.
Range of $f^{-1}$ is the domain of $f$: $y \in \mathbb{R}, y \neq 2$.