please go step by step to better understand
if 2f(x)+f(1-x)=x^2 for all real numbers, the f(x)=?
To find the function f(x) that satisfies the equation 2f(x) + f(1-x) = x^2 for all real numbers, we can go step by step:
Step 1: Substitute x = 1-x
Replacing x with (1-x), we get:
2f(1-x) + f(x) = (1-x)^2
Step 2: Simplify the equation
Expanding (1-x)^2, we have:
2f(1-x) + f(x) = 1 - 2x + x^2
Step 3: Rearrange the equation
Rearranging the equation to isolate f(x) on one side, we subtract 2f(1-x) from both sides:
f(x) = 1 - 2x + x^2 - 2f(1-x)
Step 4: Determine f(1-x)
We now need to determine the value of f(1-x) in terms of f(x). We can do this by substituting x with 1-x in the original equation:
2f(1-(1-x)) + f(x) = (1-x)^2
Simplifying further, we get:
2f(x) + f(x) = (1-x)^2
3f(x) = (1-x)^2
Step 5: Solve for f(x)
Dividing both sides by 3:
f(x) = (1-x)^2 / 3
This is the expression for the function f(x) that satisfies the given equation 2f(x) + f(1-x) = x^2 for all real numbers.