Question

Let $f\left(x\right)=x^2$ and $g\left(x\right)=2^x$. Then the solution of the equation $fog\left(x\right)=gof\left(x\right)$ is

• A. $R$
• B. $\left\{0\right\}$
• C. $\left\{0,2\right\}$
• D. none

Answer 1

The correct option is C $\left\{0,2\right\}$

$f\left(x\right)=x^2$

$fg\left(x\right)=f\left(2^x\right)={\left(2^x\right)}^2=2^{2x}$

$g\left(x\right)=2^x$

$gf\left(x\right)=g\left(x^2\right)=2^{x^2}$

Given:$fg\left(x\right)=gf\left(x\right)$

$\Rightarrow 2^{2x}=2^{x^2}$

Since bases are same we can equate the powers

$\Rightarrow 2x=x^2$

$\Rightarrow x^2-2x=0$

$\Rightarrow x(x-2)=0$

$\Rightarrow x=0,2$

When $x=0,fg\left(x\right)=gf\left(x\right)\Rightarrow 2^0=2^0=1$

When $x=2,fg\left(x\right)=gf\left(x\right)\Rightarrow 2^{2\times 2}=2^{2^2}\Rightarrow 16=16$

Hence $x=\{0,2\}$