Question

find the inverse of the function.
\\(y = x^2 + 4x + 4\\)

\\(y = \sqrt{\frac{1}{2}x + 2}\\)

\\(y = \pm\sqrt{\frac{1}{2}x + 2}\\)

\\(y = \pm\sqrt{x} - 2\\)

\\(y = \sqrt{x} - 2\\)

Explanation:

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<reasoning>

Swap variables

Using the Inverse Functions knowledge point
\[
x = y^2 + 4y + 4
\]

Factor the quadratic expression

Using the Inverse Functions knowledge point
\[
x = (y + 2)^2
\]

Solve for y

Using the Inverse Functions knowledge point
\[

$$\begin{aligned} \pm\sqrt{x} &= y + 2 \\ y &= \pm\sqrt{x} - 2 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(y = \sqrt{\frac{1}{2}x + 2}\)</mcq-option>
<mcq-option>(B) \(y = \pm\sqrt{\frac{1}{2}x + 2}\)</mcq-option>
<mcq-correct>(C) \(y = \pm\sqrt{x} - 2\)</mcq-correct>
<mcq-option>(D) \(y = \sqrt{x} - 2\)</mcq-option>
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Inverse Functions"
]
}
</post_analysis>

Answer:

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}
</pre_analysis>

<reasoning>

Swap variables

Using the Inverse Functions knowledge point
\[
x = y^2 + 4y + 4
\]

Factor the quadratic expression

Using the Inverse Functions knowledge point
\[
x = (y + 2)^2
\]

Solve for y

Using the Inverse Functions knowledge point
\[

$$\begin{aligned} \pm\sqrt{x} &= y + 2 \\ y &= \pm\sqrt{x} - 2 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(y = \sqrt{\frac{1}{2}x + 2}\)</mcq-option>
<mcq-option>(B) \(y = \pm\sqrt{\frac{1}{2}x + 2}\)</mcq-option>
<mcq-correct>(C) \(y = \pm\sqrt{x} - 2\)</mcq-correct>
<mcq-option>(D) \(y = \sqrt{x} - 2\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Inverse Functions"
]
}
</post_analysis>