Could someone help me with this calc problem?

Question

2f(x) + f(1-x) = x^2
What is f(x)?
Thanks!

Answer 1

f(x) = [x^2 + 2x - 1] / 3.

CHECK: Note that from this expression for f(x), we would get:

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

= [x^2 - 4x + 2] / 3.

Now ADD 2 f(x) + f(1 - x): it's [3x^2 + 0x + 0] / 3 = x^2. IT CHECKS!

How did i get this? If:

2 f(x) + f(1-x) = x^2, replace x by (1 - x): 2 f(1 - x) + f(x) = (1 - x)^2, or:
f(x) + 2 f(1-x) = (1 - x)^2.

Now, just call f(x) = a, and f(1 - x) = b, and solve these two preceding equations for a and b : the solutions are those given above.

As a FURTHER CHECK, a + 2 b indeed equals(1 - x)^2, as you can check for yourself. So everything used to solve this problem is confirmed.

Live long and prosper.

By the way, I just don't see how this is a CALCULUS question. It just seems to require standard algebraic manipulation, with an understanding that if one has f(x), f(1 - x) means substituting (1 - x) for x wherever it occurs.

Answer 2

just replace x with 1- x and you get other equation.
2f(1-x) + f(x) = (1-x)^2
So you will have a system with unknowns f(x) and f(1-x) and you solve the system.