Question

function: (g(x) = 2x^2 - 8)

for (x ge 0), the inverse function is (f(x) = sqrt{\frac{1}{2}x + 4})

for (x le 0), the inverse function is (d(x) = -sqrt{\frac{1}{2}x + 4})

\

$$\begin{array}{|c|c|c|} \\hline x & f(x) & d(x) \\\\ \\hline -8 & 0 & q \\\\ \\hline 0 & r & -2 \\\\ \\hline 10 & s & t \\\\ \\hline \\end{array}$$

(q = 0)
(r = 2)
(s = \\)
(t = \\)

Explanation:

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<reasoning>

Evaluate the function d(x) at x = -8 to find q

Using the Function Evaluation and Inverse Functions knowledge points
\[
q = d(-8) = -\sqrt{\frac{1}{2}(-8) + 4} = -\sqrt{-4 + 4} = 0
\]

Evaluate the function f(x) at x = 0 to find r

Using the Function Evaluation and Inverse Functions knowledge points
\[
r = f(0) = \sqrt{\frac{1}{2}(0) + 4} = \sqrt{4} = 2
\]

Evaluate the function f(x) at x = 10 to find s

Using the Function Evaluation and Inverse Functions knowledge points
\[
s = f(10) = \sqrt{\frac{1}{2}(10) + 4} = \sqrt{5 + 4} = \sqrt{9} = 3
\]

Evaluate the function d(x) at x = 10 to find t

Using the Function Evaluation and Inverse Functions knowledge points
\[
t = d(10) = -\sqrt{\frac{1}{2}(10) + 4} = -\sqrt{5 + 4} = -\sqrt{9} = -3
\]
</reasoning>

<answer>
Given the functions \(f(x) = \sqrt{\frac{1}{2}x + 4}\) and \(d(x) = -\sqrt{\frac{1}{2}x + 4}\):

• \(q = d(-8) =\) <blank>\(0\)</blank>
• \(r = f(0) =\) <blank>\(2\)</blank>
• \(s = f(10) =\) <blank>\(3\)</blank>
• \(t = d(10) =\) <blank>\(-3\)</blank>

</answer>

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Answer:

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<reasoning>

Evaluate the function d(x) at x = -8 to find q

Using the Function Evaluation and Inverse Functions knowledge points
\[
q = d(-8) = -\sqrt{\frac{1}{2}(-8) + 4} = -\sqrt{-4 + 4} = 0
\]

Evaluate the function f(x) at x = 0 to find r

Using the Function Evaluation and Inverse Functions knowledge points
\[
r = f(0) = \sqrt{\frac{1}{2}(0) + 4} = \sqrt{4} = 2
\]

Evaluate the function f(x) at x = 10 to find s

Using the Function Evaluation and Inverse Functions knowledge points
\[
s = f(10) = \sqrt{\frac{1}{2}(10) + 4} = \sqrt{5 + 4} = \sqrt{9} = 3
\]

Evaluate the function d(x) at x = 10 to find t

Using the Function Evaluation and Inverse Functions knowledge points
\[
t = d(10) = -\sqrt{\frac{1}{2}(10) + 4} = -\sqrt{5 + 4} = -\sqrt{9} = -3
\]
</reasoning>

<answer>
Given the functions \(f(x) = \sqrt{\frac{1}{2}x + 4}\) and \(d(x) = -\sqrt{\frac{1}{2}x + 4}\):

• \(q = d(-8) =\) <blank>\(0\)</blank>
• \(r = f(0) =\) <blank>\(2\)</blank>
• \(s = f(10) =\) <blank>\(3\)</blank>
• \(t = d(10) =\) <blank>\(-3\)</blank>

</answer>

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