To find the function f(x), we need to solve the given functional equation. Here is one way to approach it:
1. Substitute x-1 for x in the first term of the equation:
f((x-3)-1) + f((x+1)-1) = 2x^2 - 10x + 16
2. Simplify the expressions inside the function notation:
f(x-4) + f(x) = 2x^2 - 10x + 16
3. Substitute x+4 for x in the first term of the equation:
f((x+4)-4) + f(x) = 2x^2 - 10x + 16
4. Simplify the expressions inside the function notation:
f(0) + f(x) = 2x^2 - 10x + 16
5. Simplify the first term using the definition of the function:
f(x) + f(x) = 2x^2 - 10x + 16
6. Combine like terms on the left side of the equation:
2f(x) = 2x^2 - 10x + 16
7. Divide both sides by 2:
f(x) = x^2 - 5x + 8
Therefore, the function f(x) is f(x) = x^2 - 5x + 8.