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, $\operatorname{fog}(x)=x$ represent a singleton set.