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BasketballEinstein

Answer:

[tex]\displaystyle See\:above \\ \\ 3.\:Range: Set-Builder\:Notation → [f(x)|-8 < f(x) ≤ \frac{1}{3}] \\ Interval\:Notation → (-8, \frac{1}{3}] \\ \\ Domain: Set-Builder\:Notation → [x|x < -1\:or\:x ≥ -1] \\ Interval\:Notation → (-∞, -1) ∪ [-1, ∞) \\ \\ \\ 2.\:Range: Set-Builder\:Notation → [f(x)|4 < f(x) ≤ 4] \\ Interval\:Notation → (4, 4] \\ \\ Domain: Set-Builder\:Notation → [x|x < 2\:or\:x ≥ 2] \\ Interval\:Notation → (-∞, 2) ∪ [2, ∞) \\ \\ \\ 1.\:Range: Set-Builder\:Notation → [f(x)|-3 ≤ f(x) < 7] \\ Interval\:Notation → [-3, 7) \\ \\ Domain: Set-Builder\:Notation → [x|x ≤ -3\:or\:x > -3] \\ Interval\:Notation → (-∞, -3] ∪ (-3, ∞)[/tex]

Step-by-step explanation:

This will look complex to you, but I will explain it the best way I know how:

So, to find the range of the graph, you first need to plug in each x-value given in the inequality into the expressions right up against them, and you will get your y-value. Once both y-values are defined, you will then set them up as Set-Builder and\or Interval Notations, depending on the type of piecewise graph given to you [whether closed or opened circle].

** However, if both values are identical, then the function is considered "infinitive", but you still need to write the intervals in Set-Builder and\or Interval Notation, depending on the inequality given. On the graph, this will be one full, solid circle, but we know that there are parts of the graph where a certain interval is not included, so BE EXTREMELY CAREFUL with exercises like these [see exercise 2].

Now, the domain will be the inequalities given to you because as you can see, each exercise has the EXACT SAME x-value and unique inequalities, so just take what is there and write the intervals in Set-Builder and\or Interval Notation.

Well, I hope this helps you out alot, and as always, I am joyous to assist anyone at anytime.

*** Graphs depicted from 3 - 1

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tramserran

1. Answer: Domain: x = All Real Numbers (-∞, ∞)

                 Range: y = All Real Numbers (-∞, 7]

                 Graph: see attachment 1

1. Step-by-step explanation:

NOTE: When graphing a piecewise function, think of it as sharing the graph with 2 lines. The length of the lines are restricted by the inequality provided.

[tex]f(x)=\begin{cases}\quad x\qquad \quad ; x\le -3\\-2x+1\quad ; x>-3\end{cases}[/tex]

Graph y = x  → the line ENDS when x = -3 with a closed dot

Graph y = -2x+1 → the line STARTS when x = -3 with an open dot

There are no restrictions on x so the Domain is "All Real Numbers"

In interval notation, it is written as: (-∞, ∞)

Since both slopes are negative, there is a maximum value for y so the Range is: y < 7

In interval notation, it is written as: (-∞, 7)

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2. Answer: Domain: x = All Real Numbers (-∞, ∞)

                 Range: y = All Real Numbers (-∞, ∞)

                 Graph: see attachment 2

2. Step-by-step explanation:

[tex]f(x)=\begin{cases}\dfrac{5}{2}x-1 \quad ; x< 2\\\\\ x+2\quad ; x\ge 2\end{cases}[/tex]

Graph [tex]y=\dfrac{5}{2}x-1[/tex]  → the line ENDS when x = 2 with an open dot

Graph y = x+2 → the line STARTS when x = 2 with a closed dot

There are no restrictions on x so the Domain is "All Real Numbers"

In interval notation, it is written as: (-∞, ∞)

Since there is one slope positive and the other slope negative, there are no restrictions on y so the Range is: "All Real Numbers"

In interval notation, it is written as: (-∞, ∞)

*******************************************************************************************

3. Answer: Domain: x = All Real Numbers (-∞, ∞)

                 Range: y = All Real Numbers (-∞, 1/3]

                 Graph: see attachment 3

3. Step-by-step explanation:

[tex]f(x)=\begin{cases}-\dfrac{1}{3}x \qquad ; x\ge -1\\\\\ 2x-6\quad ; x< -1\end{cases}[/tex]

Graph [tex]y=-\dfrac{1}{3}x[/tex]  → the line STARTS when x = -1 with an open dot

Graph y = x+2 → the line ENDS when x = -1 with a closed dot

There are no restrictions on x so the Domain is "All Real Numbers"

In interval notation, it is written as: (-∞, ∞)

Since both slopes are negative, there is a maximum value for y so the Range is: y ≤ 1/3

In interval notation, it is written as: (-∞, 1/3]

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