Question

1. domain and range of \\(\\{(2,5), (4,6), (6,7), (8,8)\\}\\)

1. domain of \\(g(x) = \frac{x+2}{|x+2|}\\)

1. all relations are functions? t or f

1. \\(g(x) = -3x^3 - 4x\\) find \\(g(-1)\\)

1. \\(f(x) = 3x^2 - 3x + 4\\) find average rate what is change from 3 to 4

Explanation:

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<reasoning>

Find domain and range of the relation

\[
\text{Relation: } \{(2,5), (4,6), (6,7), (8,8)\}
\]
\[
\text{Domain (set of } x\text{-coordinates): } \{2, 4, 6, 8\}
\]
\[
\text{Range (set of } y\text{-coordinates): } \{5, 6, 7, 8\}
\]

Determine the domain of \(g(x)\)

\[
g(x) = \frac{x+2}{|x+2|}
\]
\[
|x+2|
eq 0 \implies x+2
eq 0 \implies x
eq -2
\]
\[
\text{Domain: } \{x \in \mathbb{R} \mid x
eq -2\} \text{ or } (-\infty, -2) \cup (-2, \infty)
\]

Determine if all relations are functions

\[
\text{A relation is a function if and only if each input has exactly one output.}
\]
\[
\text{Not all relations are functions (e.g., } \{(1,2), (1,3)\} \text{ is a relation but not a function).}
\]
\[
\text{Answer: F}
\]

Evaluate \(g(-1)\)

\[
g(x) = -3x^3 - 4x
\]
\[
g(-1) = -3(-1)^3 - 4(-1) = -3(-1) + 4 = 3 + 4 = 7
\]

Find the average rate of change from 3 to 4

\[
f(x) = 3x^2 - 3x + 4
\]
\[
f(3) = 3(3)^2 - 3(3) + 4 = 27 - 9 + 4 = 22
\]
\[
f(4) = 3(4)^2 - 3(4) + 4 = 48 - 12 + 4 = 40
\]
\[
\text{Average Rate of Change} = \frac{f(4) - f(3)}{4 - 3} = \frac{40 - 22}{1} = 18
\]
</reasoning>

<answer>

Question 1

• Domain: \(\{2, 4, 6, 8\}\)
• Range: \(\{5, 6, 7, 8\}\)

Question 2

• Domain: \(\{x \in \mathbb{R} \mid x

eq -2\}\) or \((-\infty, -2) \cup (-2, \infty)\)

Question 3

• Answer: F (False)

Question 4

• \(g(-1) = 7\)

Question 5

• Average Rate of Change: \(18\)

</answer>

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Answer:

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<reasoning>

Find domain and range of the relation

\[
\text{Relation: } \{(2,5), (4,6), (6,7), (8,8)\}
\]
\[
\text{Domain (set of } x\text{-coordinates): } \{2, 4, 6, 8\}
\]
\[
\text{Range (set of } y\text{-coordinates): } \{5, 6, 7, 8\}
\]

Determine the domain of \(g(x)\)

\[
g(x) = \frac{x+2}{|x+2|}
\]
\[
|x+2|
eq 0 \implies x+2
eq 0 \implies x
eq -2
\]
\[
\text{Domain: } \{x \in \mathbb{R} \mid x
eq -2\} \text{ or } (-\infty, -2) \cup (-2, \infty)
\]

Determine if all relations are functions

\[
\text{A relation is a function if and only if each input has exactly one output.}
\]
\[
\text{Not all relations are functions (e.g., } \{(1,2), (1,3)\} \text{ is a relation but not a function).}
\]
\[
\text{Answer: F}
\]

Evaluate \(g(-1)\)

\[
g(x) = -3x^3 - 4x
\]
\[
g(-1) = -3(-1)^3 - 4(-1) = -3(-1) + 4 = 3 + 4 = 7
\]

Find the average rate of change from 3 to 4

\[
f(x) = 3x^2 - 3x + 4
\]
\[
f(3) = 3(3)^2 - 3(3) + 4 = 27 - 9 + 4 = 22
\]
\[
f(4) = 3(4)^2 - 3(4) + 4 = 48 - 12 + 4 = 40
\]
\[
\text{Average Rate of Change} = \frac{f(4) - f(3)}{4 - 3} = \frac{40 - 22}{1} = 18
\]
</reasoning>

<answer>

Question 1

• Domain: \(\{2, 4, 6, 8\}\)
• Range: \(\{5, 6, 7, 8\}\)

Question 2

• Domain: \(\{x \in \mathbb{R} \mid x

eq -2\}\) or \((-\infty, -2) \cup (-2, \infty)\)

Question 3

• Answer: F (False)

Question 4

• \(g(-1) = 7\)

Question 5

• Average Rate of Change: \(18\)

</answer>

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