SOLUTION: Given a function f of a real variable which satisfies the function defined by x*f(2-x)-2*f(x)=x^2+1 for all real x, determine f(x) in terms of x.

SOLUTION: Given a function f of a real variable which satisfies the function defined by xf(2-x)-2f(x)=x^2+1 for all real x, determine f(x) in terms of x.

Question 1165361: Given a function f of a real variable which satisfies the function defined by x*f(2-x)-2*f(x)=x^2+1 for all real x, determine f(x) in terms of x.
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
Let's make

When the variable is x, the equation is
, which we write in abbreviated form as
(1)

When the variable is 2-x, the equation is
, which we want to simplify as
, and
, and abbreviate as
(2)

In (1) and (2), we have a system of equations on and :
. Let's solve for .

Multiplying (2) times we get ,
which we simplify to
,
and add it to (1) to get
,

is not zero for any real number,
so we can divide by that expression to solve for

The function is
.
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