Question

If $f\left(x\right)=\{\begin{array}{c}x+1,ifx\le 1\\ 5-x^{2}ifx>1\end{array},g\left(x\right)=\{\begin{array}{c}xifx\le 1\\ 2-xifx>1\end{array}$

Number of negative integral solutions of $g\left(f\right(x\left)\right)+2=0$ are

• A. $0$
• B. $3$
• C. $1$
• D. $2$

Answer 1

The correct option is C $1$

$f\left(x\right)=\{\begin{array}{c}x+1,ifx\le 1\\ 5-x^{2}ifx>1\end{array},$

$g\left(x\right)=\{\begin{array}{c}xifx\le 1\\ 2-xifx>1\end{array}$

$g\left(f\right(x\left)\right)+2=0,g\left(f\right(x\left)\right)=-2,g\left(f\right(x\left)\right)=\{\begin{array}{c}x+1ifx\le 0\\ 1-xif1\ge x>0\\ x^2-3if2>x>1\\ 5-x^{2}ifx\ge 2\end{array}Forx<0tobeasolution,x+1=-2⟶1solution$

Hence, option C.