Question

determine whether the graph is that of a function by using the vertical-line test. in either case, use the graph to find the following.

(a) the domain and range
(b) the intercepts, if any
(c) any symmetry with respect to the x-axis, y-axis, or the origin

Explanation:

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"Vertical Line Test",
"Domain and Range",
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"Graph Symmetry"
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</pre_analysis>

<reasoning>

Determine if the graph represents a function

\[
\text{Every vertical line intersects the graph at most once} \implies \text{It is a function.}
\]

Find the domain and range

\[

$$\begin{aligned} &\text{Domain: } [-5, 0] \\ &\text{Range: } [-3, 2] \end{aligned}$$

\]

Find the intercepts and symmetry

\[

$$\begin{aligned} &\text{x-intercept: } (-4, 0) \\ &\text{y-intercept: } (0, -3) \\ &\text{Symmetry: None} \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

The graph represents a function because it passes the vertical-line test.

Question 2

The domain is \([-5, 0]\) and the range is \([-3, 2]\).

Question 3

The x-intercept is \((-4, 0)\) and the y-intercept is \((0, -3)\).

Question 4

The graph has no symmetry with respect to the x-axis, y-axis, or the origin.
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Algebra",
"Graph Analysis"
]
}
</post_analysis>

Answer:

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"step_cot",
"explain_then_answer",
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],
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"concepts_used": [
"Vertical Line Test",
"Domain and Range",
"Intercepts",
"Graph Symmetry"
],
"new_concepts": [],
"current_concepts": [
"Vertical Line Test",
"Domain and Range",
"Intercepts",
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}
</pre_analysis>

<reasoning>

Determine if the graph represents a function

\[
\text{Every vertical line intersects the graph at most once} \implies \text{It is a function.}
\]

Find the domain and range

\[

$$\begin{aligned} &\text{Domain: } [-5, 0] \\ &\text{Range: } [-3, 2] \end{aligned}$$

\]

Find the intercepts and symmetry

\[

$$\begin{aligned} &\text{x-intercept: } (-4, 0) \\ &\text{y-intercept: } (0, -3) \\ &\text{Symmetry: None} \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

The graph represents a function because it passes the vertical-line test.

Question 2

The domain is \([-5, 0]\) and the range is \([-3, 2]\).

Question 3

The x-intercept is \((-4, 0)\) and the y-intercept is \((0, -3)\).

Question 4

The graph has no symmetry with respect to the x-axis, y-axis, or the origin.
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Algebra",
"Graph Analysis"
]
}
</post_analysis>