Question

If $f:R\to R,g:R\to R,h:R\to R$ be three functions given by $f\left(x\right)=x^2-1,g\left(x\right)=\sqrt{x^2+1},$ $h\left(x\right)=\{\begin{array}{cc}0,& x\le 0\\ x,& x>0\end{array}$ then $\left(hofog\right)\left(x\right)=$

• A. $\{\begin{array}{cc}{x}^2+1,& x>0\\ x^2-1,& x\le 0\end{array}$
• B. $\{\begin{array}{cc}{x}^2+1,& x>0\\ 0,& x\le 0\end{array}$
• C. $\{\begin{array}{cc}-x^2,& x<0\\ 0,& x=0\\ x^2& x>0\end{array}$
• D. $\{\begin{array}{cc}{x}^2,& x\ne 0\\ 0,& x=0\end{array}$

Answer 1

The correct option is D $\{\begin{array}{cc}{x}^2,& x\ne 0\\ 0,& x=0\end{array}$

$\left(hofog\right)\left(x\right)=h\left(f\right(g\left(x\right)\left)\right)$ $=h(f\left(\sqrt{x^2+1)}\right)$
$=h(x^2+1-1)$
$=h\left(x^2\right)$
$=\{\begin{array}{cc}{x}^2,& x\ne 0\\ 0,& x=0\end{array}$