Question

If $f\left(x\right)=2^{100}x+1,g\left(x\right)=3^{100}x+1$, then the set of real numbers $x$ such that $f\left\{g\right(x\left)\right\}=x$ is

• A. empty
• B. a singleton
• C. a finite set with more than one element
• D. infinite

Answer 1

Answer: (B)

a singleton

Given $f\left(x\right)=2^{100}x+1,g\left(x\right)=3^{100}x+1$

Now $fog\left(x\right)=x$

$\Rightarrow$ $f(3^{100}.x+1)=x$
$\Rightarrow$ $2^{100}(3^{100}.x+1)+1=x$
$\Rightarrow$ $6^{100}.x+2^{100}+1=x$

$\Rightarrow$ $x(1-6^{100})=(1+2^{100})$

$\Rightarrow$ $x=\frac{1+2^{100}}{1-6^{100}}$

Hence $fog\left(x\right)=x$ represent a singleton set.