Question

If $f\left(x\right)=x^2$ and $g\left(x\right)=2^x$, find solution set of fog(x)= gof(x).

Answer 1

$f\left(x\right)=x^{2}g\left(x\right)=2^{x}f\left(g\right(x)={\left\{g\right(x\left)\right\}}^2\Rightarrow f(g\left(x\right)={\left\{2^x\right\}}^2=2^{2x}g\left(f\right(x\left)\right)=2^{f\left(x\right)}\Rightarrow g\left(f\right(x\left)\right)=2^{x^2}gof\left(x\right)=fog\left(x\right)2^{x^2}=2^{2x}\Rightarrow x^2=2x{x}^2-2x=0x(x-2)=0x=0,2$