Question

Let
$f(x)=2^{100}x+1$,
$g(x)=3^{100}x+1$.
Then the set of real numbers $x$ such that $f(g(x))=x$ is

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

Answer 1

Answer: (B)

Given, $f(x)=2^{100}\cdot x+1$
and $g(x)=3^{100}\cdot x+1$
Now, $f\{g(x)\}=x$
$\Rightarrow f\left(3^{100}\cdot x+1\right)=x$
$\Rightarrow 2^{100}\left\{3^{100}\cdot x+1\right\}+1=x$
$\Rightarrow 6^{100}\cdot x+2^{100}+1=x$
$\Rightarrow x\left(1-6^{100}\right)=\left(1+2^{100}\right)$
$\Rightarrow x=\frac{1+2^{100}}{1-6^{100}}$
Hence, $\mathrm{fog}(x)=x$ represent a singleton set.