Question

If $\text{sgn}\left(y\right)$ denotes the signum function of $y,$ then the range of $f\left(x\right)=\text{sgn}(x^2+4x+5)$ is

• A. $[1,\infty )$
• B. $\{-1,0,1\}$
• C. $\left\{1\right\}$
• D. $\{0,1\}$

Answer 1

The correct option is C $\left\{1\right\}$

Clearly, the domain of $f$ is $R$
$x^2+4x+5=(x+2{)}^2+1\ge 1$ for all $x\in R$
We know that,
$\text{sgn}\left(x\right)=\{\begin{array}{cc}-1,& x<0\\ 0,& x=0\\ 1,& x>0\end{array}$

$\therefore f\left(x\right)=\text{sgn}(x^2+4x+5)=1$
Hence, range of $f\left(x\right)$ is $\left\{1\right\}.$