Let's solve for the values of \(q\), \(r\), \(s\), and \(t\) step-by-step using the given functions and the table.
1. Calculate \(q\):
[tex]\[
q = d(-8)
\][/tex]
Given \(d(x) = -\sqrt{\frac{1}{2} x + 4}\),
[tex]\[
d(-8) = -\sqrt{\frac{1}{2} (-8) + 4} = -\sqrt{-4 + 4} = -\sqrt{0} = -0
\][/tex]
Thus,
[tex]\[
q = -0 \quad \text{(or just 0, but for consistency we'll use -0)}
\][/tex]
2. Calculate \(r\):
[tex]\[
r = f(0)
\][/tex]
Given \(f(x) = \sqrt{\frac{1}{2} x + 4}\),
[tex]\[
f(0) = \sqrt{\frac{1}{2} (0) + 4} = \sqrt{0 + 4} = \sqrt{4} = 2
\][/tex]
Thus,
[tex]\[
r = 2
\][/tex]
3. Calculate \(s\):
[tex]\[
s = f(10)
\][/tex]
Given \(f(x) = \sqrt{\frac{1}{2} x + 4}\),
[tex]\[
f(10) = \sqrt{\frac{1}{2} (10) + 4} = \sqrt{5 + 4} = \sqrt{9} = 3
\][/tex]
Thus,
[tex]\[
s = 3
\][/tex]
4. Calculate \(t\):
[tex]\[
t = d(10)
\][/tex]
Given \(d(x) = -\sqrt{\frac{1}{2} x + 4}\),
[tex]\[
d(10) = -\sqrt{\frac{1}{2} (10) + 4} = -\sqrt{5 + 4} = -\sqrt{9} = -3
\][/tex]
Thus,
[tex]\[
t = -3
\][/tex]
Collating these results, we get:
[tex]\[
\begin{array}{l}
q = -0 \\
r = 2 \\
s = 3 \\
t = -3
\end{array}
\][/tex]
So, the completed table is:
[tex]\[
\begin{array}{|c|c|c|}
\hline x & f(x) & d(x) \\
\hline -8 & 0 & -0 \\
\hline 0 & 2 & -2 \\
\hline 10 & 3 & -3 \\
\hline
\end{array}
\][/tex]
