Question

Let $f\left(x\right)=x^3$ and $g\left(x\right)=3^x$. The values of $A$ such that $g\left[f\right(A\left)\right]=f\left[g\right(A\left)\right]$ are

• A. $0.2$
• B. $1,3$
• C. $0,\pm 3$
• D. $1,\pm 2$
• E. $0,\pm \sqrt{3}$

Answer 1

The correct option is E $0,\pm \sqrt{3}$

Given, $f\left(x\right)=x^2$ and $g\left(x\right)=3^x$
$g\left[f\right(A\left)\right]=g\left(A^3\right)=3^{A^3}$
and $f\left[g\right(A\left)\right]=g\left(3^A\right)={\left(3^A\right)}^3=3^{3A}$
since $g\left[f\right(A\left)\right]=f\left[g\right(A\left)\right]$
$\Rightarrow A^3-3A\Rightarrow A(A^2-3)=0$
$\Rightarrow A=\pm \sqrt{3}$
Hence, the values are $0,\pm \sqrt{3}$