if g(x-1) =1-g(x)/ 8 and g(1) =3 find the value of g(2)

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.

if g(x-1) = 1 - g(x)/8 and g(1) = 3, find the value of g(2)

Learn how to solve the functional equation g(x-1) = (1 - g(x)) / 8 with the initial condition g(1) = 3. This problem explores substitution in functional equations and is suitable for students in Grades 9-12.

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