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Question: Answered & Verified by Expert
If $\mathrm{g}(\mathrm{x})=\frac{1}{\mathrm{f}(\mathrm{x})}$ and $\mathrm{f}(\mathrm{x})=\mathrm{x}, \mathrm{x} \neq 0$, then which one of the
following is correct $\quad[2015-I I]$
MathematicsFunctionsNDANDA 2015 (Phase 2)
Options:
  • A $\quad \operatorname{f}(\mathrm{f}(\mathrm{f}(\mathrm{g}(\mathrm{g}(\mathrm{f}(\mathrm{x}))))))=\mathrm{g}(\mathrm{g}(\mathrm{f}(\mathrm{g}(\mathrm{f}(\mathrm{x})))))$
  • B $\quad \mathrm{f}(\mathrm{f}(\mathrm{g}(3(\mathrm{~g}(\mathrm{f}(\mathrm{x}))))))=\mathrm{g}(\mathrm{g}(\mathrm{f}(\mathrm{g}(\mathrm{f}(\mathrm{x}))))))$
  • C $\mathrm{f}(\mathrm{g}(\mathrm{f}(\mathrm{g}(\mathrm{g}(\mathrm{f}(\mathrm{g}(\mathrm{x}))))))=\mathrm{g}(\mathrm{g}(\mathrm{f}(\mathrm{g}(\mathrm{f}(\mathrm{x}))))))$
  • D $\quad \mathrm{f}(\mathrm{f}(\mathrm{f}(\mathrm{f}(\mathrm{f}(\mathrm{f}(\mathrm{x}))))))=\mathrm{f}(\mathrm{f}(\mathrm{f}(\mathrm{g}(\mathrm{f}(\mathrm{x})))))$
Solution:
2007 Upvotes Verified Answer
The correct answer is: $\quad \mathrm{f}(\mathrm{f}(\mathrm{g}(3(\mathrm{~g}(\mathrm{f}(\mathrm{x}))))))=\mathrm{g}(\mathrm{g}(\mathrm{f}(\mathrm{g}(\mathrm{f}(\mathrm{x}))))))$
$f(x)=x$ $\mathrm{g}(\mathrm{x})=1 / \mathrm{x}$
Putting these values in the options, only (b) is correct.

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