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Let's analyze the given piecewise-defined function [tex]\( f(x) \)[/tex]:
[tex]\[ f(x)=\begin{cases} 3x + 1, & \text{if } -9 < x \leq -2 \\ -5, & \text{if } -2 < x \leq 1 \\ x - 6, & \text{if } 1 < x < 7 \end{cases} \][/tex]
### Domain of the Function
The domain [tex]\( D \)[/tex] of the function [tex]\( f(x) \)[/tex] is the union of all the intervals for which [tex]\( f(x) \)[/tex] is defined:
[tex]\[ D: (-9, 7) \][/tex]
### Analyzing Each Piece of the Function
1. First Piece: [tex]\( f(x) = 3x + 1 \)[/tex] for [tex]\(-9 < x \leq -2 \)[/tex]
- Range calculation:
- When [tex]\( x = -9 \)[/tex]:
[tex]\[ f(-9) = 3(-9) + 1 = -27 + 1 = -26 \][/tex]
- When [tex]\( x = -2 \)[/tex]:
[tex]\[ f(-2) = 3(-2) + 1 = -6 + 1 = -5 \][/tex]
- So, the range for this interval is:
[tex]\[ R_1: (-26, -5] \][/tex]
- Behavior:
- Since the coefficient of [tex]\( x \)[/tex] (which is 3) is positive, the function [tex]\( 3x + 1 \)[/tex] is increasing on this interval.
2. Second Piece: [tex]\( f(x) = -5 \)[/tex] for [tex]\(-2 < x \leq 1 \)[/tex]
- Range calculation:
- For any value of [tex]\( x \)[/tex] in the interval [tex]\(-2 < x \leq 1\)[/tex]:
[tex]\[ f(x) = -5 \][/tex]
- So, the range for this interval is:
[tex]\[ R_2: \{-5\} \][/tex]
- Behavior:
- The function [tex]\( f(x) = -5 \)[/tex] is constant on this interval.
3. Third Piece: [tex]\( f(x) = x - 6 \)[/tex] for [tex]\(1 < x < 7 \)[/tex]
- Range calculation:
- When [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 1 - 6 = -5 \][/tex]
- When [tex]\( x = 7 \)[/tex]:
[tex]\[ f(7) = 7 - 6 = 1 \][/tex]
- So, the range for this interval is:
[tex]\[ R_3: (-5, 1) \][/tex]
- Behavior:
- Since the coefficient of [tex]\( x \)[/tex] (which is 1) is positive, the function [tex]\( x - 6 \)[/tex] is increasing on this interval.
### Summarizing Behavior and Ranges
- For [tex]\(-9 < x \leq -2\)[/tex], [tex]\( f(x) = 3x + 1 \)[/tex] is increasing, with range [tex]\((-26, -5]\)[/tex].
- For [tex]\(-2 < x \leq 1\)[/tex], [tex]\( f(x) = -5 \)[/tex] is constant, with range [tex]\(\{-5\}\)[/tex].
- For [tex]\(1 < x < 7\)[/tex], [tex]\( f(x) = x - 6 \)[/tex] is increasing, with range [tex]\((-5, 1)\)[/tex].
### Domain and Range
- Domain:
[tex]\[ D: (-9, 7) \][/tex]
- Range:
[tex]\[ R = R_1 \cup R_2 \cup R_3 = (-26, -5] \cup \{-5\} \cup (-5, 1) = [-26, 1) \][/tex]
### Graphing the Function
To graph the function, plot each piece on the respective intervals:
1. Plot the line [tex]\( 3x + 1 \)[/tex] for [tex]\( -9 < x \leq -2 \)[/tex]. This line will start at [tex]\( (-9, -26) \)[/tex] and end at [tex]\( (-2, -5) \)[/tex]. The endpoint at [tex]\( (-2, -5) \)[/tex] should be a solid dot since [tex]\( x = -2 \)[/tex] is included in this piece.
2. Plot the horizontal line [tex]\( y = -5 \)[/tex] for [tex]\( -2 < x \leq 1 \)[/tex]. This line is constant at [tex]\( y = -5 \)[/tex] and will have a solid dot at [tex]\( (1, -5) \)[/tex] since [tex]\( x = 1 \)[/tex] is included in this piece.
3. Plot the line [tex]\( x - 6 \)[/tex] for [tex]\( 1 < x < 7 \)[/tex]. This line will start at [tex]\( (1, -5) \)[/tex] and end at [tex]\( (7, 1) \)[/tex]. Both endpoints should be plotted as open circles since neither [tex]\( x = 1 \)[/tex] nor [tex]\( x = 7 \)[/tex] are included in this piece.
The graph provides a visual understanding of the function's domain, range, and behavior on each interval.
[tex]\[ f(x)=\begin{cases} 3x + 1, & \text{if } -9 < x \leq -2 \\ -5, & \text{if } -2 < x \leq 1 \\ x - 6, & \text{if } 1 < x < 7 \end{cases} \][/tex]
### Domain of the Function
The domain [tex]\( D \)[/tex] of the function [tex]\( f(x) \)[/tex] is the union of all the intervals for which [tex]\( f(x) \)[/tex] is defined:
[tex]\[ D: (-9, 7) \][/tex]
### Analyzing Each Piece of the Function
1. First Piece: [tex]\( f(x) = 3x + 1 \)[/tex] for [tex]\(-9 < x \leq -2 \)[/tex]
- Range calculation:
- When [tex]\( x = -9 \)[/tex]:
[tex]\[ f(-9) = 3(-9) + 1 = -27 + 1 = -26 \][/tex]
- When [tex]\( x = -2 \)[/tex]:
[tex]\[ f(-2) = 3(-2) + 1 = -6 + 1 = -5 \][/tex]
- So, the range for this interval is:
[tex]\[ R_1: (-26, -5] \][/tex]
- Behavior:
- Since the coefficient of [tex]\( x \)[/tex] (which is 3) is positive, the function [tex]\( 3x + 1 \)[/tex] is increasing on this interval.
2. Second Piece: [tex]\( f(x) = -5 \)[/tex] for [tex]\(-2 < x \leq 1 \)[/tex]
- Range calculation:
- For any value of [tex]\( x \)[/tex] in the interval [tex]\(-2 < x \leq 1\)[/tex]:
[tex]\[ f(x) = -5 \][/tex]
- So, the range for this interval is:
[tex]\[ R_2: \{-5\} \][/tex]
- Behavior:
- The function [tex]\( f(x) = -5 \)[/tex] is constant on this interval.
3. Third Piece: [tex]\( f(x) = x - 6 \)[/tex] for [tex]\(1 < x < 7 \)[/tex]
- Range calculation:
- When [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 1 - 6 = -5 \][/tex]
- When [tex]\( x = 7 \)[/tex]:
[tex]\[ f(7) = 7 - 6 = 1 \][/tex]
- So, the range for this interval is:
[tex]\[ R_3: (-5, 1) \][/tex]
- Behavior:
- Since the coefficient of [tex]\( x \)[/tex] (which is 1) is positive, the function [tex]\( x - 6 \)[/tex] is increasing on this interval.
### Summarizing Behavior and Ranges
- For [tex]\(-9 < x \leq -2\)[/tex], [tex]\( f(x) = 3x + 1 \)[/tex] is increasing, with range [tex]\((-26, -5]\)[/tex].
- For [tex]\(-2 < x \leq 1\)[/tex], [tex]\( f(x) = -5 \)[/tex] is constant, with range [tex]\(\{-5\}\)[/tex].
- For [tex]\(1 < x < 7\)[/tex], [tex]\( f(x) = x - 6 \)[/tex] is increasing, with range [tex]\((-5, 1)\)[/tex].
### Domain and Range
- Domain:
[tex]\[ D: (-9, 7) \][/tex]
- Range:
[tex]\[ R = R_1 \cup R_2 \cup R_3 = (-26, -5] \cup \{-5\} \cup (-5, 1) = [-26, 1) \][/tex]
### Graphing the Function
To graph the function, plot each piece on the respective intervals:
1. Plot the line [tex]\( 3x + 1 \)[/tex] for [tex]\( -9 < x \leq -2 \)[/tex]. This line will start at [tex]\( (-9, -26) \)[/tex] and end at [tex]\( (-2, -5) \)[/tex]. The endpoint at [tex]\( (-2, -5) \)[/tex] should be a solid dot since [tex]\( x = -2 \)[/tex] is included in this piece.
2. Plot the horizontal line [tex]\( y = -5 \)[/tex] for [tex]\( -2 < x \leq 1 \)[/tex]. This line is constant at [tex]\( y = -5 \)[/tex] and will have a solid dot at [tex]\( (1, -5) \)[/tex] since [tex]\( x = 1 \)[/tex] is included in this piece.
3. Plot the line [tex]\( x - 6 \)[/tex] for [tex]\( 1 < x < 7 \)[/tex]. This line will start at [tex]\( (1, -5) \)[/tex] and end at [tex]\( (7, 1) \)[/tex]. Both endpoints should be plotted as open circles since neither [tex]\( x = 1 \)[/tex] nor [tex]\( x = 7 \)[/tex] are included in this piece.
The graph provides a visual understanding of the function's domain, range, and behavior on each interval.
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