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1998

JEE Main 1998 Mathematics question on Sets Relations and Functions

If g{f(x)}=∣sin⁡x∣g\{f(x)\} = |\sin x| and f{g(x)}=(sin⁡x)2f\{g(x)\} = (\sin \sqrt{x})^2, then
  1. A.
    f(x)=sin⁡2x, g(x)=xf(x) = \sin^2 x,\ g(x) = \sqrt{x}
  2. B.
    f(x)=sin⁡x, g(x)=∣x∣f(x) = \sin x,\ g(x) = |x|
  3. C.
    f(x)=x2, g(x)=sin⁡xf(x) = x^2,\ g(x) = \sin \sqrt{x}
  4. D.
    ff and gg cannot be determined
Show answer and solution

Correct option: A

Solution

With f(x) = sin²x and g(x) = √x:

g{f(x)} = g(sin²x) = √(sin²x) = |sin x| ✓

f{g(x)} = f(√x) = (sin√x)² = sin²(√x) ✓

Both compositions match exactly, confirming option 1.

[1998]

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