Exploring the Multiplicative Property of Functions: f(x, y) f(x)f(y) and Non-Constant Solutions

In the realm of mathematical functions, the multiplicative property f(x, y) f(x)f(y) has drawn considerable attention from mathematicians. This functional equation suggests that the function f exhibits a unique behavior when applied to two variables. A significant question arises: is it necessary for f(x) to be a constant function?

Basic Cases

To understand this property, let's start with some basic cases:

Case 1: Setting y 1

If we set y 1, the equation transforms to:

f(x, 1) f(x)f(1)

This simplifies to:

f(x) f(x)f(1)

Assuming f(x) ≠ 0, we can divide both sides by f(x) to obtain:

1 f(1)

Thus, f(1) 1.

Case 2: Evaluating f(0)

Next, let's evaluate f(0). If we set x 0 and y 0, the equation becomes:

f(0, 0) f(0)f(0)

This simplifies to:

f(0) f(0)2

This implies:

f(0) - f(0)2 0

Thus, f(0) 0 or f(0) 1.

If f(0) 0, for any x:

0 f(x)f(0) f(x)0

Which is always true and does not give us a contradiction.

If f(0) 1, we need to explore further.

Exploring Non-Constant Solutions

A common solution to this type of functional equation is the exponential function. If we assume f(x) a^{g(x)} for some function g, then:

a^{g(x, y)} a^{g(x) g(y)}

This implies:

g(x y) g(x) g(y)

The general solutions to this are of the form g(x) k log x for some constant k, leading to:

f(x) x^k for some constant k.

If k 0, we get:

f(x) 1 a constant function.

If k ≠ 0, we get non-constant functions.

Conclusion

Therefore, the function f(x, y) f(x)f(y) can be satisfied by functions such as f(x) c^k where c is a constant and k is any real number. This means that while f(x) can be a constant function like f(x) 1, it is not necessary for f(x) to be a constant. Non-constant solutions like f(x) x^k for k ≠ 0 also exist.

In summary, it is not necessary that f(x) is a constant function, and non-constant solutions exist.

Additional Examples

Well, clearly not. The identity function fx x is a function whose domain at a given point is exactly the same as its range. The function defined by f: R → R, f(x) x is said to be an identity function and the graph of this function is a straight line passing through the origin and inclined at an angle 45° to the positive direction of the X-axis. If f is an identity function, then f(x) x, fy y, and then finally f(x, y) xy.

From the given equation, we can write:

f(x, y) xy f(x)f(y)

This example demonstrates that fx x^n for any n also works. Additionally, there could be an infinite number of non-constant functions that satisfy the equation f(x, y) f(x)f(y).