\$-\\infty < x < 1\$ \$-\\infty < x < \\infty\$ \\(-\\infty < x < 2…

QUESTION IMAGE

$\$-\\infty < x < 1\$ \$-\\infty < x < \\infty\$ \\(-\\infty < x < 2…$

Question

\$-\infty < x < 1\$
\$-\infty < x < \infty\$
\$-\infty < x < 2\$
\$1 < x < \infty\$

Explanation:

Response

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Increasing Intervals",
"Quadratic Functions",
"Vertex of Parabola"
],
"new_concepts": [],
"current_concepts": [
"Increasing Intervals",
"Quadratic Functions",
"Vertex of Parabola"
]
}
</pre_analysis>

Identify the vertex of the parabola

The given graph is a downward-opening parabola. The peak or vertex of this parabola is located at the point $(1, 1)$.

Determine the interval where the function is increasing

A function is increasing where its graph rises from left to right. For this downward-opening parabola, the graph rises for all $x$-values to the left of the vertex's $x$-coordinate.
\[
x < 1 \implies -\infty < x < 1
\]

Match with the given options

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

<answer>
<mcq-correct>(A) $-\infty < x < 1$</mcq-correct>
<mcq-option>(B) $-\infty < x < \infty$</mcq-option>
<mcq-option>(C) $-\infty < x < 2$</mcq-option>
<mcq-option>(D) $1 < x < \infty$</mcq-option>
</answer>

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

Answer: