Question

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\\(1 < x < \infty\\)
\\(1 < x < 1\\)
\\(-\infty < x < 1\\)
\\(-1 < x < \infty\\)

Explanation:

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<reasoning>

Identify the vertex of the parabola

The graph shows a downward-opening parabola. The highest point (vertex) of this parabola is located at the coordinates \((1, 1)\).

Determine the interval where the function is increasing

A function is increasing on an interval if the \(y\)-values increase as the \(x\)-values increase from left to right. For this downward-opening parabola, the curve rises from left to right up to the \(x\)-coordinate of the vertex.
\[
x < 1 \implies -\infty < x < 1
\]

Match with the given options

The interval of increase is \(-\infty < x < 1\), which corresponds to the third option.
</reasoning>

<answer>
<mcq-option>(A) \(1 < x < \infty\)</mcq-option>
<mcq-option>(B) \(1 < x < 1\)</mcq-option>
<mcq-correct>(C) \(-\infty < x < 1\)</mcq-correct>
<mcq-option>(D) \(-1 < x < \infty\)</mcq-option>
</answer>

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"question_type": "Multiple Choice",
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Answer:

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<reasoning>

Identify the vertex of the parabola

The graph shows a downward-opening parabola. The highest point (vertex) of this parabola is located at the coordinates \((1, 1)\).

Determine the interval where the function is increasing

A function is increasing on an interval if the \(y\)-values increase as the \(x\)-values increase from left to right. For this downward-opening parabola, the curve rises from left to right up to the \(x\)-coordinate of the vertex.
\[
x < 1 \implies -\infty < x < 1
\]

Match with the given options

The interval of increase is \(-\infty < x < 1\), which corresponds to the third option.
</reasoning>

<answer>
<mcq-option>(A) \(1 < x < \infty\)</mcq-option>
<mcq-option>(B) \(1 < x < 1\)</mcq-option>
<mcq-correct>(C) \(-\infty < x < 1\)</mcq-correct>
<mcq-option>(D) \(-1 < x < \infty\)</mcq-option>
</answer>

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