Question

Let $f$,$g$ and $h$ be the function defined by $f\left(x\right)=\frac{1}{1+\log x}$ where $x\in R^{+}-\left\{0.1\right\}$, $g\left(x\right)=x^2$, $x\in R$ and $h\left(x\right)=1+x^2$, $x\in R$ then $\left(hofog\right)\left(x\right)$ is (here codomain of $\left(hofog\right)\left(x\right)$ be $R$)

• A. one one and onto
• B. many one and onto
• C. one one and into
• D. many one and into

Answer 1

The correct option is D many one and into

$g\left(x\right)=x^2$
$\left(fog\right)\left(x\right)=\frac{1}{1+\log x^2}$
$\left(hofog\right)\left(x\right)=1+{\left(\frac{1}{1+\log x^2}\right)}^2$
$\left(hofog\right)\left(x\right)=1+{\left(\frac{1}{1+2\log |x|}\right)}^2$
clearly we can see that range of $\left(hofog\right)\left(x\right)$ wiil be greater than $1$ so it is into
And it gives same value on $x=R^{+}$ and $x=R^{-}$
So it is many one