## Precalculus (6th Edition) Blitzer

Domain of $f\left( x \right):\left( -\infty ,\infty \right)$ , Range of $f\left( x \right):\left( -\infty ,9 \right]$
The graph is a parabola. Since $f\left( x \right)$ is a polynomial, it exists for every value of $x$ on the real axis. Therefore, the domain of $f\left( x \right)=-{{x}^{2}}-4x+5$ is $\left( -\infty ,\infty \right)$. Since, \begin{align} & -{{x}^{2}}-4x+5=-\left( {{x}^{2}}+4x \right)+5 \\ & =-\left( {{x}^{2}}+4x+4-4 \right)+5 \\ & =-\left( {{\left( x+2 \right)}^{2}}-4 \right)+5 \\ & =9-{{\left( x+2 \right)}^{2}} \end{align} Also, $\begin{matrix} {{\left( x+2 \right)}^{2}}\ge 0 \\ -{{\left( x+2 \right)}^{2}}\le 0 \\ 9-{{\left( x+2 \right)}^{2}}\le 9 \\ f\left( x \right)\le 9 \\ \end{matrix}$ $\text{Then, }f\left( x \right)\in \left( -\infty ,9 \right]$ So, the range of the function is $\left( -\infty ,9 \right]$.