Question

Let $f(x)$ be a function satisfying $f(x)f(y) = f(xy)$ for all real $x, y$. If $f(2) = 4$, then what is the value of $f\left(\frac{1}{2}\right)$?

Accepted Answer

B. Since $f(x)f(y) = f(xy)$: For $x = 2, y = 1$, $f(2)f(1) = f(2) \implies 4f(1) = 4 \implies f(1) = 1$. For $x = 2, y = \frac{1}{2}$: $$f(2)f\left(\frac{1}{2}ight) = f\left(2 	imes \frac{1}{2}ight) = f(1)$$ $$4 	imes f\left(\frac{1}{2}ight) = 1 \implies f\left(\frac{1}{2}ight) = \frac{1}{4}$$.

Functional Equation

Explanation

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