Let $f(x)=x^{2}$ and $g(x)=2^{x}$. Then, the solution set of the equation $f o g(x)=g o f(x)$ is

Question. Let $f(x)=x^{2}$ and $g(x)=2^{x}$. Then, the solution set of the equation $f o g(x)=g o f(x)$ is

Solution:
Since $(f o g)(x)=(g o f)(x)$,
$f(g(x))=g(f(x))$
$\Rightarrow f\left(2^{x}\right)=g\left(x^{2}\right)$
$\Rightarrow\left(2^{x}\right)^{2}=2^{x^{2}}$
$\Rightarrow 2^{2 x}=2^{x^{2}}$
$\Rightarrow x^{2}=2 x$
$\Rightarrow x^{2}-2 x=0$
$\Rightarrow x(x-2)=0$
$\Rightarrow x=0,2$
$\Rightarrow x \in\{0,2\}$
So, the answer is (c).

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