Answer
$g(-6)=\sqrt{11}
,\\\\
g(3)=\text{does not exist}
,\\\\
g(6)=\sqrt{11}
,\\\\
g(13)=12$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Substitute the given function value in $
g(x)=\sqrt{x^2-25}
.$
$\bf{\text{Solution Details:}}$
If $
x=-6
,$ then
\begin{array}{l}\require{cancel}
g(x)=\sqrt{x^2-25}
\\\\
g(-6)=\sqrt{(-6)^2-25}
\\\\
g(-6)=\sqrt{36-25}
\\\\
g(-6)=\sqrt{11}
.\end{array}
If $
x=3
,$ then
\begin{array}{l}\require{cancel}
g(x)=\sqrt{x^2-25}
\\\\
g(3)=\sqrt{(3)^2-25}
\\\\
g(3)=\sqrt{9-25}
\\\\
g(3)=\sqrt{-16}
\text{ (not a real number)}
.\end{array}
If $
x=6
,$ then
\begin{array}{l}\require{cancel}
g(x)=\sqrt{x^2-25}
\\\\
g(6)=\sqrt{(6)^2-25}
\\\\
g(6)=\sqrt{36-25}
\\\\
g(6)=\sqrt{11}
.\end{array}
If $
x=13
,$ then
\begin{array}{l}\require{cancel}
g(x)=\sqrt{x^2-25}
\\\\
g(13)=\sqrt{(13)^2-25}
\\\\
g(13)=\sqrt{169-25}
\\\\
g(13)=\sqrt{144}
\\\\
g(13)=12
.\end{array}
Hence,
\begin{array}{l}\require{cancel}
g(-6)=\sqrt{11}
,\\\\
g(3)=\text{does not exist}
,\\\\
g(6)=\sqrt{11}
,\\\\
g(13)=12
.\end{array}
