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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Functions

NDA 2024 · Mathematics · Functions

Let f(x) f(y)=f(x y) for all real x, y . If f(2)=4 , then what is...

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

Options

  1. A14\frac{1}{4}
  2. B12\frac{1}{2}
  3. C11
  4. D44

Correct answer

A. 14\frac{1}{4}

Step-by-step solution

Given f(x)=f(y)f(x)=f(y) Let f(x)=\mathrmxn\mathrm{f}(\mathrm{x})=\mathrmx^{\mathrm{n}} and f(y)=\mathrmyn\mathrm{f}(\mathrm{y})=\mathrmy^{\mathrm{n}}. f(xy)=(xy)nf(2)=4⇒4=(2)n⇒n=2f(xy)=(xy)2f(12)=(12)2=14\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}
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