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 \(f\left( \frac12 \right)\)?

• A. 0
• B. \(\frac14\)
• C. \(\frac12\)
• D. 1
• E. cannot be determined

Hint:

Functional equations often allow finding special values by substituting convenient \(x, y\) pairs.

Answer 1

The Correct Option is B

From \(f(x)f(y) = f(xy)\), put \(x = 2\) and \(y = \frac12\): \[ f(2)f\left(\frac12\right) = f(1) \] Also, \(f(1)f(1) = f(1)\) ⇒ \(f(1)^2 = f(1)\) ⇒ \(f(1) = 0\) or \(f(1) = 1\). If \(f(1) = 0\), then \(f(2) = 0\) contradicts \(f(2) = 4\). So \(f(1) = 1\). Thus: \[ 4 \cdot f\left(\frac12\right) = 1 \quad\Rightarrow\quad f\left(\frac12\right) = \frac14 \]