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