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:

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

$$ f(x) = 3x^2 - 3x + 4 $$

$$ f(3) = 3(3)^2 - 3(3) + 4 = 27 - 9 + 4 = 22 $$

$$ f(4) = 3(4)^2 - 3(4) + 4 = 48 - 12 + 4 = 40 $$

$$ \text{Average Rate of Change} = \frac{f(4) - f(3)}{4 - 3} = \frac{40 - 22}{1} = 18 $$

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: \(18\)