Question

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

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

Answer 1

The correct option is C

$\left\{0,2\right\}$

Explanation for the correct option:

Determining the solution set for $\left(fog\right(x\left)\right)=\left(gof\right(x\left)\right)$.

Given that: $f\left(x\right)=x^2$, $g\left(x\right)=2^x$

$$\begin{gathered} \Rightarrow \left(fog\right(x\left)\right)=\left(gof\right(x\left)\right)\left[\mathrm{Given}\right] \\ \Rightarrow {\left[g\left(x\right)\right]}^2=2^{\left[f\left(x\right)\right]} \\ \Rightarrow {\left(2^x\right)}^2=2^{x^2} \\ \Rightarrow 2^{2x}=2^{x^2} \\ \Rightarrow 2x=x^2 \end{gathered}$$

$$\begin{gathered} \Rightarrow x^2-2x=0 \\ \Rightarrow x=0,2 \end{gathered}$$

Therefore, we can say that the solution set of $\left(fog\right(x\left)\right)=\left(gof\right(x\left)\right)$ will be $\left\{0,2\right\}$

Hence, the correct option is (C).