Question

Please give me The solution to such questions f(x-3)+f(x+1)=2x^2-10x+16
f(x)=??

Answer 1

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.