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If $f(x)=e^{x}$ and $g(x)=\log e^{x}$, then which of the following is true?
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Verified Answer
The correct answer is:
$f\{g(x)\}=g\{f(x)\}$
We have, $f(x)=e^{x}$ and $g(x)=\log e^{x}$
Now, $f(g(x))=f\left(\log e^{x}\right)=e^{\log e^{x}}=e^{x}$
and $g(f(x))=g\left(e^{x}\right)=\log e^{e^{x}}=e^{x} \log e=e^{x}$
Hence, $f(g(x))=g(f(x))$.
Now, $f(g(x))=f\left(\log e^{x}\right)=e^{\log e^{x}}=e^{x}$
and $g(f(x))=g\left(e^{x}\right)=\log e^{e^{x}}=e^{x} \log e=e^{x}$
Hence, $f(g(x))=g(f(x))$.
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