Question

Let $f ( x )=\ln \left( e ^{ x }+1\right)$ and $g _1( x )= f ( x )$ and $g _{ n +1}( x )= f \left( g _{ n }( x )\right) \forall n \geq 1$. Then the number of real roots of the equation $g _{10}( x )= x$ is

• A. 8
• B. 4
• C. 2
• D. 0

Answer 1

Answer: (D)

$g _1( x )=\ln \left( e ^{ x }+1\right) $
$g _2( x )= f \left(\ln \left( e ^{ x }+1\right)\right)=\ln \left( e ^{ x }+2\right) $
$\text { so } g _3( x )=\ln \left( e ^{ x }+3\right) \ldots . ., g _{10}( x )=\ln \left( e ^{ x }+10\right) $