SI: f(x+3) =3g(3x+2) -(x.x)-1   ; g(x+5) =2f(3x-4)-4x+2; calcular: f(1+g(1+f(5)))

To solve the problem, we need to determine \( f(1 + g(1 + f(5))) \). The given functions are:

\[
f(x+3) = 3g(3x+2) - x^2 - 1
\]

\[
g(x+5) = 2f(3x-4) - 4x + 2
\]

### Step 1: Find \( f(5) \)
Let's substitute \( x = 2 \) into the first equation to find \( f(5) \):

\[
f(2+3) = f(5) = 3g(3(2) + 2) - 2^2 - 1
\]

\[
f(5) = 3g(6 + 2) - 4 - 1 = 3g(8) - 5
\]

We now have \( f(5) = 3g(8) - 5 \).

### Step 2: Find \( g(1 + f(5)) \)
Substitute \( x = 1 + f(5) \) into the function \( g(x+5) \):

\[
g(1 + f(5) + 5) = g(6 + f(5)) 
\]

\[
g(6 + f(5)) = 2f(3(1 + f(5)) - 4) - 4(1 + f(5)) + 2
\]

Now, substitute \( f(5) = 3g(8) - 5 \) into this equation. For simplicity, let's denote \( f(5) \) by \( A = 3g(8) - 5 \):

\[
g(6 + A) = 2f(3(1 + A) - 4) - 4(1 + A) + 2
\]

\[
g(6 + A) = 2f(3 + 3A - 4) - 4 - 4A + 2
\]

Simplify:

\[
g(6 + A) = 2f(3A - 1) - 4A - 2
\]

### Step 3: Substitute into \( f(1 + g(1 + f(5))) \)

Now, substitute \( g(6 + A) \) into \( f(1 + g(1 + f(5))) \):

\[
f(1 + g(1 + A)) = f(1 + g(6 + A))
\]

Using \( g(6 + A) = 2f(3A - 1) - 4A - 2 \), we then substitute it into \( f(x) \):

\[
f(1 + (2f(3A - 1) - 4A - 2))
\]

This expression is complex and typically would require more information or solving the system of equations formed by these functions explicitly for specific values to find \( f(1 + g(1 + f(5))) \).

### Final Step:
Given the complexity, we'd either continue simplifying or potentially substitute values back into the original equations to solve iteratively. Without additional simplifications or numerical methods, this cannot be simplified further analytically.

If you need more detailed steps or have any questions, let me know!

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### 5 Related Questions
1. How can we solve systems of functional equations?
2. What are typical methods for solving equations involving compositions of functions?
3. How does substitution work in solving complex functional equations?
4. What techniques can simplify nested function evaluations?
5. How can we approximate solutions for non-linear function equations?

### Tip:
When dealing with complex functional equations, substituting specific values and simplifying step by step can help in understanding the pattern or structure of the solution.

Explore how to solve complex functional equations involving composition of functions step-by-step. This problem involves determining values of f(x) and g(x) using given functional relationships and substitution methods. Suitable for advanced high school or college-level mathematics.

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