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 = \\)
\\(r = \\)
\\(s = \\)
\\(t = \\)

Explanation:

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<reasoning>

Identify the given functions

We are given two inverse functions:
\[
f(x) = \sqrt{\frac{1}{2}x + 4}
\]
\[
d(x) = -\sqrt{\frac{1}{2}x + 4}
\]

Calculate the value of q

The table shows that when \(x = -8\), \(d(x) = q\).
We substitute \(x = -8\) into the formula for \(d(x)\):
\[
q = d(-8) = -\sqrt{\frac{1}{2}(-8) + 4}
\]
\[
q = -\sqrt{-4 + 4} = -\sqrt{0} = 0
\]

Calculate the value of r

The table shows that when \(x = 0\), \(f(x) = r\).
We substitute \(x = 0\) into the formula for \(f(x)\):
\[
r = f(0) = \sqrt{\frac{1}{2}(0) + 4}
\]
\[
r = \sqrt{4} = 2
\]

Calculate the value of s

The table shows that when \(x = 10\), \(f(x) = s\).
We substitute \(x = 10\) into the formula for \(f(x)\):
\[
s = f(10) = \sqrt{\frac{1}{2}(10) + 4}
\]
\[
s = \sqrt{5 + 4} = \sqrt{9} = 3
\]

Calculate the value of t

The table shows that when \(x = 10\), \(d(x) = t\).
We substitute \(x = 10\) into the formula for \(d(x)\):
\[
t = d(10) = -\sqrt{\frac{1}{2}(10) + 4}
\]
\[
t = -\sqrt{5 + 4} = -\sqrt{9} = -3
\]
</reasoning>

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

\(q =\) <blank>0</blank>

\(r =\) <blank>2</blank>

\(s =\) <blank>3</blank>

\(t =\) <blank>-3</blank>
</answer>

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"question_type": "Fill-in-the-blank",
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"Algebra",
"Inverse Functions"
]
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</post_analysis>

Answer:

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<reasoning>

Identify the given functions

We are given two inverse functions:
\[
f(x) = \sqrt{\frac{1}{2}x + 4}
\]
\[
d(x) = -\sqrt{\frac{1}{2}x + 4}
\]

Calculate the value of q

The table shows that when \(x = -8\), \(d(x) = q\).
We substitute \(x = -8\) into the formula for \(d(x)\):
\[
q = d(-8) = -\sqrt{\frac{1}{2}(-8) + 4}
\]
\[
q = -\sqrt{-4 + 4} = -\sqrt{0} = 0
\]

Calculate the value of r

The table shows that when \(x = 0\), \(f(x) = r\).
We substitute \(x = 0\) into the formula for \(f(x)\):
\[
r = f(0) = \sqrt{\frac{1}{2}(0) + 4}
\]
\[
r = \sqrt{4} = 2
\]

Calculate the value of s

The table shows that when \(x = 10\), \(f(x) = s\).
We substitute \(x = 10\) into the formula for \(f(x)\):
\[
s = f(10) = \sqrt{\frac{1}{2}(10) + 4}
\]
\[
s = \sqrt{5 + 4} = \sqrt{9} = 3
\]

Calculate the value of t

The table shows that when \(x = 10\), \(d(x) = t\).
We substitute \(x = 10\) into the formula for \(d(x)\):
\[
t = d(10) = -\sqrt{\frac{1}{2}(10) + 4}
\]
\[
t = -\sqrt{5 + 4} = -\sqrt{9} = -3
\]
</reasoning>

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

\(q =\) <blank>0</blank>

\(r =\) <blank>2</blank>

\(s =\) <blank>3</blank>

\(t =\) <blank>-3</blank>
</answer>

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"question_type": "Fill-in-the-blank",
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"Mathematics",
"Algebra",
"Inverse Functions"
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