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Question
Let \[f\left( x \right) = x^2 and g\left( x \right) = 2^x\] Then, the solution set of the equation
Options
R
{0}
{0, 2}
none of these
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Solution
\[\text{Since}\left( \text{fog} \right)\left( x \right) = \left( \text{gof} \right)\left( x \right), \]
\[ f\left( g\left( x \right) \right) = g\left( f\left( x \right) \right)\]
\[ \Rightarrow f\left( 2^x \right) = g\left( x^2 \right)\]
\[ \Rightarrow \left( 2^x \right)^2 = 2^{x^2} \]
\[ \Rightarrow 2^{2x} = 2^{x^2} \]
\[ \Rightarrow x^2 = 2x\]
\[ \Rightarrow x^2 - 2x = 0\]
\[ \Rightarrow x\left( x - 2 \right) = 0\]
\[ \Rightarrow x = 0, 2\]
\[ \Rightarrow x \in \left\{ 0, 2 \right\}\]
So, the answer is (c) .
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