Question

Let $f\left(x\right)=x^2\mathrm{and}g\left(x\right)=2^x$. Then, the solution set of the equation $fog\left(x\right)=gof\left(x\right)$ is
(a) R
(b) {0}
(c) {0, 2}
(d) none of these

Answer 1

$$\begin{gathered} \text{Since}\left(fog\right)\left(x\right)=\left(gof\right)\left(x\right), \\ f\left(g\left(x\right)\right)=g\left(f\left(x\right)\right) \\ \Rightarrow f\left(2^x\right)=g\left(x^2\right) \\ \Rightarrow {\left(2^x\right)}^2=2^{x^2} \\ \Rightarrow 2^{2x}=2^{x^2} \\ \Rightarrow x^2=2x \\ \Rightarrow x^2-2x=0 \\ \Rightarrow x\left(x-2\right)=0 \\ \Rightarrow x=0,2 \\ \Rightarrow x\in \left\{0,2\right\} \end{gathered}$$
So, the answer is (c).