The problem defines a function $g(x) = \frac{3x}{1-x}$. We are asked to find two things: (i) The value of $g(2)$. (ii) The image of 1 under $g^{-1}$, which is equivalent to finding $g^{-1}(1)$.

1. Problem Description

The problem defines a function $g(x) = \frac{3x}{1-x}$. We are asked to find two things:

(i) The value of $g(2)$.

(ii) The image of 1 under $g^{-1}$, which is equivalent to finding $g^{-1}(1)$.

2. Solution Steps

(i) To find $g(2)$, we substitute $x=2$ into the expression for $g(x)$:

$g(2) = \frac{3(2)}{1-2} = \frac{6}{-1} = -6$.

(ii) To find $g^{-1}(1)$, we need to find the value of $x$ such that $g(x) = 1$. In other words, we need to solve the equation $\frac{3x}{1-x} = 1$ for $x$.

$3x = 1-x$

$3x + x = 1$

$4x = 1$

$x = \frac{1}{4}$

Therefore, $g^{-1}(1) = \frac{1}{4}$.

3. Final Answer

(i) $g(2) = -6$

(ii) The image of 1 under $g^{-1}$ is $\frac{1}{4}$.