If f(x+1)+f(x+3)=f(x)+f(x+2)f(x+1)+f(x+3)=f(x)+f(x+2), what's ff?

Answer 1

First we must know what f(x) is equal to... starting from right to the left, and ignoring the messy middle expression, notice we have:

f(x) + f(x+2) = f(x+1) + f(x+3) .subtract f(x+2) from both sides ==> f(x) =f(x+1) + f(x +3) - f(x+2)

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

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

Answer 2

There are four unknowns but only three equations....

Let y = f(x)

y1 = f(x+1)

y2 = f(x+2)

y3 = f(x+3)

The equations are:

y1 + y3 = y + y2*y1 + y3

y1 + y3 = y + y2

y + y2*y1 + y3 = y + y2

Decide which one is to be the free variable, solve each one so that the free var is ALONE on one side;

then solve the 3x3, treating the free var as a fixed constant.

If 2 of the equations are the same you will then have TWO (2) free vars.