We are given the functional equation:

$g(x - 1) = \frac{1 - g(x)}{8}$

and the condition $g(1) = 3$.

We need to find the value of $g(2)$.

Step 1: Use $g(1) = 3$ to find $g(0)$

Let $x = 1$ in the given equation: $g(1 - 1) = \frac{1 - g(1)}{8}$ This simplifies to: $g(0) = \frac{1 - 3}{8}$ $g(0) = \frac{-2}{8} = -\frac{1}{4}$

Step 2: Use $g(2)$ to express $g(1)$

Next, let $x = 2$ in the original equation: $g(2 - 1) = \frac{1 - g(2)}{8}$ This simplifies to: $g(1) = \frac{1 - g(2)}{8}$ Substitute $g(1) = 3$ into this equation: $3 = \frac{1 - g(2)}{8}$ Multiply both sides by 8: $24 = 1 - g(2)$ Now, solve for $g(2)$: $g(2) = 1 - 24 = -23$

Thus, $g(2) = -23$.

Final Answer:

The value of $g(2)$ is $\boxed{-23}$.

Would you like further details or clarifications? Here are some related questions for deeper understanding:

1. How do we use the concept of functional equations in other types of problems?
2. What happens if we are given a different initial condition for $g(x)$?
3. Can you generalize this method to solve any function defined recursively?
4. How can you check if a function like this has an inverse?
5. How do properties of transformations help solve functional equations like this?

Tip: When solving functional equations, using given values to substitute and simplify step-by-step can help uncover unknown values efficiently.