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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Functions

NDA 2024 · Mathematics · Functions

Paragraph: Let f ^ g(x)= ^2 x and g ^ f(x)=| x| . Question: Which...

Paragraph: Let f∘g(x)=cos⁡2xf {}^{\circ} g(x)=\cos ^2 \sqrt{x} and g∘f(x)=∣cos⁡x∣g {}^{\circ} f(x)=|\cos x|.
Question: Which one of the following is f(x)f(x) ?

Options

  1. Acos⁡x\cos x
  2. Bcos⁡x2\cos x^2
  3. Ccos⁡2x\cos ^2 x
  4. Dcos⁡∣x∣\cos |x|

Correct answer

C. cos⁡2x\cos ^2 x

Step-by-step solution

Given f∘g(x)=cos⁡2xf {}^{\circ} g(x)=\cos ^2 \sqrt{x}, go f(x)=∣cos⁡x∣f(x)=|\cos x| If we take option (c) f(x)=cos⁡2xf(x)=\cos ^2 x, then g(x)=xg(x)=\sqrt{x} fog (x)=cos⁡2x and g of (x)=cos⁡2x=∣cos⁡x∣∴f(x)=cos⁡2x\begin{aligned} & \text{ fog }(x)=\cos ^2 \sqrt{x} \text{ and } g \text{ of }(x)=\sqrt{\cos ^2 x}=|\cos x| \\ & \therefore f(x)=\cos ^2 x \end{aligned}$$ so option (c) is correct.
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