Question

Let the functions $ f,\,\,\,g,\,\,\,h $ are defined from the set of real numbers $ R $ to $ R $ such that $ f(x)={{x}^{2}}-1,\,\,g(x)=\sqrt{({{x}^{2}}+1)} $ and $ h(x)=\left\{ \begin{matrix} 0,if & x\le 0 \\ x,if & x\ge 0 \\ \end{matrix} \right. $ , then $ ho(fog)(x) $ is defined by

• A. $ x $
• B. $ {{x}^{2}} $
• C. $ 0 $
• D. None of these

Answer 1

Answer: (B)

Given that, $ f(x)={{x}^{2}}-1,\,\,g(x)=\sqrt{({{x}^{2}}+1)} $ and $ h(x)=\left\{ \begin{matrix} 0,if & x<0 \\ x,if & x\ge 0 \\ \end{matrix} \right. $
$ \therefore $ $ ho(fog)(x)=hof\{g(x)\} $
$ =hof\{\sqrt{({{x}^{2}}+1)}\} $
$ =h\{{{(\sqrt{{{x}^{2}}+1})}^{2}}-1\} $
$ =h\{{{x}^{2}}+1-1\} $
$ =h\{{{x}^{2}}\} $
$ ={{x}^{2}} $