Answer
Domain of $f\left( x \right):\left( -\infty ,\infty \right)$ , Range of $f\left( x \right):\left( -\infty ,9 \right]$
Work Step by Step
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 $\left( -\infty ,9 \right]$.
