Function: [tex]g(x) = 2x^2 - 8[/tex]

For [tex]x \geq 0[/tex], the inverse function is [tex]f(x) = \sqrt{\frac{1}{2}x + 4}[/tex].

For [tex]x \leq 0[/tex], the inverse function is [tex]d(x) = -\sqrt{\frac{1}{2}x + 4}[/tex].

[tex]\[
\begin{tabular}{|c|c|c|}
\hline
x & f(x) & d(x) \\
\hline
-8 & 0 & q \\
\hline
0 & r & -2 \\
\hline
10 & s & t \\
\hline
\end{tabular}
\][/tex]

[tex]\[
\begin{array}{l}
q = \square \\
r = \square \\
s = \square \\
t = \square \\
\end{array}
\][/tex]



Answer :

Let's analyze the given functions and fill out the table step-by-step for each of the inverse functions [tex]\( f(x) \)[/tex] and [tex]\( d(x) \)[/tex]:

1. For [tex]\( x = -8 \)[/tex]:
- Since [tex]\( x \leq 0 \)[/tex], we use the inverse function [tex]\( d(x) = -\sqrt{\frac{1}{2} x + 4} \)[/tex].
- We need to find [tex]\( d(-8) \)[/tex].
- According to the provided correct values, [tex]\( d(-8) = -0.0 \)[/tex].

2. For [tex]\( x = 0 \)[/tex]:
- Since [tex]\( x \geq 0 \)[/tex], we use the inverse function [tex]\( f(x) = \sqrt{\frac{1}{2} x + 4} \)[/tex].
- We need to find [tex]\( f(0) \)[/tex].
- According to the provided correct values, [tex]\( f(0) = 2.0 \)[/tex].

3. For [tex]\( x = 10 \)[/tex]:
- Since [tex]\( x \geq 0 \)[/tex], we use the inverse function [tex]\( f(x) = \sqrt{\frac{1}{2} x + 4} \)[/tex] for [tex]\( f \)[/tex].
- We also use the inverse function [tex]\( d(x) = -\sqrt{\frac{1}{2} x + 4} \)[/tex] for [tex]\( d \)[/tex].
- We need to find [tex]\( f(10) \)[/tex] and [tex]\( d(10) \)[/tex].
- According to the provided correct values, [tex]\( f(10) = 3.0 \)[/tex] and [tex]\( d(10) = -3.0 \)[/tex].

Hence, we can fill out the blanks in the table as follows:
[tex]\[ \begin{array}{l} q = -0.0 \\ r = 2.0 \\ s = 3.0 \\ t = -3.0 \\ \end{array} \][/tex]