Q80 / 120 · NDA 2024 (Phase 2) · Mathematics

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

  • Correct answer
    A
    $\frac{1}{4}$
  • B
    $\frac{1}{2}$
  • C
    $1$
  • D
    $4$

Solution

Given $f(x)=f(y)$ Let $\mathrm{f}(\mathrm{x})=\mathrm{x}^{\mathrm{n}}$ and $\mathrm{f}(\mathrm{y})=\mathrm{y}^{\mathrm{n}}$. $\begin{aligned} & \mathrm{f}(\mathrm{xy})=(\mathrm{x} \mathrm{y})^{\mathrm{n}} \\ & \mathrm{f}(2)=4 \Rightarrow 4=(2)^{\mathrm{n}} \Rightarrow \mathrm{n}=2 \\ & \mathrm{f}(\mathrm{xy})=(\mathrm{xy})^2 \\ & \mathrm{f}\left(\frac{1}{2}\right)=\left(\frac{1}{2}\right)^2=\frac{1}{4} \end{aligned}$

Hence, the correct answer is option: 'A'.

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