To determine the values of [tex]\( q \)[/tex], [tex]\( r \)[/tex], [tex]\( s \)[/tex], and [tex]\( t \)[/tex], we need to correctly evaluate the inverse functions [tex]\( f(x) \)[/tex] and [tex]\( d(x) \)[/tex] at the given [tex]\(x\)[/tex] values as outlined in the problem statement.
1. Evaluate [tex]\( d(x) \)[/tex] at [tex]\( x = -8 \)[/tex] to find [tex]\( q \)[/tex]:
Since [tex]\( x = -8 \leq 0 \)[/tex], we use [tex]\( d(x) = -\sqrt{\frac{1}{2} x + 4} \)[/tex]:
[tex]\[
d(-8) = -\sqrt{\frac{1}{2} (-8) + 4} = -\sqrt{-4 + 4} = -\sqrt{0} = -0.0
\][/tex]
So, [tex]\( q = -0.0 \)[/tex].
2. Evaluate [tex]\( f(x) \)[/tex] at [tex]\( x = 0 \)[/tex] to find [tex]\( r \)[/tex]:
Since [tex]\( x = 0 \geq 0 \)[/tex], we use [tex]\( f(x) = \sqrt{\frac{1}{2} x + 4} \)[/tex]:
[tex]\[
f(0) = \sqrt{\frac{1}{2} (0) + 4} = \sqrt{0 + 4} = \sqrt{4} = 2.0
\][/tex]
So, [tex]\( r = 2.0 \)[/tex].
3. Evaluate [tex]\( f(x) \)[/tex] at [tex]\( x = 10 \)[/tex] to find [tex]\( s \)[/tex]:
Since [tex]\( x = 10 \geq 0 \)[/tex], we use [tex]\( f(x) = \sqrt{\frac{1}{2} x + 4} \)[/tex]:
[tex]\[
f(10) = \sqrt{\frac{1}{2} (10) + 4} = \sqrt{5 + 4} = \sqrt{9} = 3.0
\][/tex]
So, [tex]\( s = 3.0 \)[/tex].
4. Evaluate [tex]\( d(x) \)[/tex] at [tex]\( x = 10 \)[/tex] to find [tex]\( t \)[/tex]:
Although [tex]\( x = 10 \geq 0 \)[/tex], the problem explicitly specifies evaluating [tex]\( d(x) \)[/tex], therefore:
[tex]\[
d(10) = -\sqrt{\frac{1}{2} (10) + 4} = -\sqrt{5 + 4} = -\sqrt{9} = -3.0
\][/tex]
So, [tex]\( t = -3.0 \)[/tex].
In summary:
[tex]\[
\begin{array}{l}
q = -0.0 \\
r = 2.0 \\
s = 3.0 \\
t = -3.0 \\
\end{array}
\][/tex]
So the completed table with the calculated values is:
[tex]\[
\begin{array}{|c|c|c|}
\hline
x & f(x) & d(x) \\
\hline
-8 & 0 & -0.0 \\
\hline
0 & 2.0 & -2 \\
\hline
10 & 3.0 & -3.0 \\
\hline
\end{array}
\][/tex]
And the values:
[tex]\[
q = -0.0 \quad r = 2.0 \quad s = 3.0 \quad t = -3.0
\][/tex]
