Analysis of the Functional Equation fxy fx * fy - 2fxy
The functional equation you provided is:
fxy fx * fy - 2fxy
To analyze this equation, we can explore specific forms of solutions. A common approach is to test simple functions such as linear functions or polynomials.
Testing for Linear Solutions
Assume that fx ax b for some constants a and b. We will compute each side of the equation and match them.
Left-hand side
fxy axy b
Right-hand side
fx * fy - 2fxy (ax b)(ay b) - 2axy
Simplifying this, we get:
(ax b)(ay b) - 2axy a^2xy abx aby b^2 - 2axy a^2xy abx aby b^2 - 2axy a^2xy abx aby b^2 - 2axy
This simplifies to:
a^2xy abx aby b^2 - 2axy
Matching the left-hand side and the right-hand side:
axy b a^2xy abx aby b^2 - 2axy
Step-by-Step Analysis
For the equation to hold true for all x and y, the constant terms and the coefficients of xy must match.
Constant term:
b b^2
This implies that b 0.
Coefficient of xy term:
1a a^2 - 2a
Therefore, we have:
0 a^2 - 3a
This simplifies to:
a(a - 3) 0
Hence, a 0 or a 3.
Considering a 0:
When a 0, the function becomes:
fx 0
This is a valid solution.
Considering a 3:
If a 3, the equation becomes:
3x 0 0
This implies:
3x 0
This is not a valid function for all x.
Therefore, the only valid solution is:
fx 0
Calculus-free Approach
Let's present a calculus-free solution to show that the only function satisfying the functional relation is fequiv 0.
Step 0: Introductory Step
Letting x y 1 yields:
f2 2f1 - 2f1 0
Step 1: Substitution with Arbitrary x
For x arbitrary, taking y 1 yields:
fx1 fx * f1 - 2fx f1 - fx
Step 2: Substitution with x1
In the expression above, substitute x by x1, reaching:
fx2 f1 - fx1
Step 3: Combining Steps 1 and 2
Combining Step 1 and Step 2 gives:
fx2 fx
Step 4: Applying the Original Relation
In the original relation namely fxy fx * fy - 2fxy, take y 2 and combine with Steps 0 and 3:
fx2 fx * f2 - 2f2x
This simplifies to:
f2x 0
Since f2x 0 holds for all x, the proof is complete.
Thus, the only solution to the functional equation is:
fequiv 0