Solving for Real-Valued Functions Satisfying a Specific Functional Equation

The problem in question is to find all real-valued functions that satisfy the functional equation:

fx · fy (f(?(xy)))2

This is a complex and intriguing problem that requires a thorough and insightful analysis. Let's dive in!

Understanding the Functional Equation

First, observe that when y 0, the equation simplifies to:

fx · f0 (f(0))2

From this, we can consider two cases based on the value of fx0.

If fx0 0, then:

(f(0))2 0

This implies:

boxed{ fx 0 } quad forall x in mathbb{R}

In this case, fx has no zeros for any real value of x. This is because if fx 0 for some x, then from the equation:

fx·f(-x) (f(0))2 0

This would imply:

fx0 0

which contradicts our assumption that fx0 ≠ 0.

Additionally, the equation shows that either:

fx 0 quad forall x in mathbb{R}

or:

fx 0 quad forall x in mathbb{R}

Because if fx · f(-x) 0, then fx and f(-x) must be of the same sign.

Let f0 a. Then, we have:

(?xf)2 afx

Which implies:

f2x a(fx/a)

Through induction, we find that:

fnx a([fx/a]n

This holds true for all natural numbers n. Extending this to integers, rational numbers, and finally real numbers, we define:

frx a([fx/a]r)

Substituting this back into the original functional equation, we find:

fx aex

for constants a and b.

Thus, the function fx that satisfies the given functional equation is:

boxed{ fx aex }