Suppose for all integers $$ x $$, there are two functions $$ f $$ and $$ g $$ such that
$$
f(x) + f(x-1) - 1 = 0
$$
and
$$
g(x) = x^2.
$$
If $$ f(x^2 - x) = 5 $$, then the value of the sum $$ f(g(5)) + g(f(5)) $$ is:

Question

Suppose for all integers $$ x $$, there are two functions $$ f $$ and $$ g $$ such that  
$$
f(x) + f(x-1) - 1 = 0
$$  
and  
$$
g(x) = x^2.
$$  
If $$ f(x^2 - x) = 5 $$, then the value of the sum $$ f(g(5)) + g(f(5)) $$ is:

Accepted Answer

1. **Understand the Functional Equation**:  
   The given equation is:  
   $$
   f(x) + f(x-1) - 1 = 0.
   $$  
   Rearrange to express $$ f(x) $$:  
   $$
   f(x) = 1 - f(x-1).
   $$

2. **Determine Behavior of $$ f(x) $$**:  
   Using the recurrence relation, we observe that $$ f(x) $$ alternates between two values depending on whether $$ x $$ is even or odd:  
   - For even $$ x $$: $$ f(x) = 5 $$.  
   - For odd $$ x $$: $$ f(x) = -4 $$.

3. **Given Condition**:  
   It is given that $$ f(x^2 - x) = 5 $$.  
   For this to hold, $$ x^2 - x $$ must be even (as $$ f(	ext{even}) = 5 $$). This is true because:  
   - $$ x^2 - x = x(x-1) $$, which is always a product of two consecutive integers and hence even.

4. **Evaluate $$ g(5) + g(f(5)) $$**:  
   - From the definition of $$ g(x) = x^2 $$:  
     $$
     g(5) = 5^2 = 25.
     $$  
   - Determine $$ f(5) $$: Since 5 is odd, $$ f(5) = -4 $$.  
     Substitute into $$ g(f(5)) = g(-4) = (-4)^2 = 16. $$

5. **Final Calculation**:  
   Add the results:  
   $$
   f(g(5)) + g(f(5)) = f(25) + 16.
   $$  
   Since 25 is odd, $$ f(25) = -4 $$:  
   $$
   (-4) + 16 = 12.
   $$

Thus, the value of the sum is **12**.