5

5. [Maximum mark: 8]
   A function $f$ is defined by $f(x) = \frac{2(x+3)}{3(x+2)}$, where $x \in \mathbb{R}, x \neq -2$. The graph $y = f(x)$ is shown below.

(a) Write down the equation of the horizontal asymptote.

Consider $g(x) = mx + 1$, where $m \in \mathbb{R}, m \neq 0$.

(b) (i) Write down the number of solutions to $f(x) = g(x)$ for $m > 0$.

(ii) Determine the value of $m$ such that $f(x) = g(x)$ has only one solution for $x$.

(iii) Determine the range of values for $m$, where $f(x) = g(x)$ has two solutions for $x \ge 0$.