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To graph the piecewise-defined function and analyze its characteristics, let's break it down step by step.
### Step 1: Define the Function
We have a piecewise function given by:
[tex]\[ f(x) = \begin{cases} 2|x + 4| - 6 & \text{for } -6 \leq x \leq -2 \\ \frac{5}{2} x + 5 & \text{for } -2 < x \leq 0 \\ \frac{-2}{3}|x - 3| + 20 & \text{for } 0 < x \leq 6 \end{cases} \][/tex]
### Step 2: Determine the Domain
The domain of the function is the set of all possible input values [tex]\( x \)[/tex]. According to the piecewise definition, the function is defined from [tex]\( x = -6 \)[/tex] to [tex]\( x = 6 \)[/tex].
Thus, the domain is:
[tex]\[ \text{Domain: } [-6, 6] \][/tex]
### Step 3: Determine the Range
The range of a function is the set of all possible output values [tex]\( f(x) \)[/tex].
From the given answer:
[tex]\[ \text{Range: } (-2, 18.0) \][/tex]
This indicates the lowest possible value of [tex]\( f(x) \)[/tex] is -2 and the highest is 18.0.
### Step 4: Analyze Each Piece of the Function
For [tex]\( -6 \leq x \leq -2 \)[/tex]:
[tex]\[ f(x) = 2|x + 4| - 6 \][/tex]
- [tex]\( |x + 4| \)[/tex] creates a V-shaped graph centered at [tex]\( x = -4 \)[/tex].
- The term [tex]\(-6\)[/tex] shifts this graph downward by 6 units.
- The function decreases from [tex]\( x = -6 \)[/tex] to [tex]\( x = -4 \)[/tex] and then increases from [tex]\( x = -4 \)[/tex] to [tex]\( x = -2 \)[/tex].
For [tex]\( -2 < x \leq 0 \)[/tex]:
[tex]\[ f(x) = \frac{5}{2} x + 5 \][/tex]
- This is a linear function with a slope of [tex]\(\frac{5}{2}\)[/tex] and y-intercept of 5.
- Since the slope is positive, the function is increasing throughout the interval [tex]\( -2 < x \leq 0 \)[/tex].
For [tex]\( 0 < x \leq 6 \)[/tex]:
[tex]\[ f(x) = \frac{-2}{3}|x - 3| + 20 \][/tex]
- [tex]\( |x - 3| \)[/tex] creates a V-shaped graph centered at [tex]\( x = 3 \)[/tex].
- The [tex]\(\frac{-2}{3}\)[/tex] coefficient inverts the V-shape, making it an upside-down V, and the [tex]\( +20 \)[/tex] shifts it upward by 20 units.
- The function increases from [tex]\( x = 0 \)[/tex] to [tex]\( x = 3 \)[/tex] and then decreases from [tex]\( x = 3 \)[/tex] to [tex]\( x = 6 \)[/tex].
### Step 5: Identify Increasing, Constant, or Decreasing Intervals
Based on the given answer:
- Decreasing intervals are from [tex]\( -6 \)[/tex] to [tex]\( -4.006 \)[/tex] and from [tex]\( 3.009 \)[/tex] to [tex]\( 6 \)[/tex].
- Constant intervals do not exist.
- Increasing intervals are from [tex]\( -4.006 \)[/tex] to [tex]\( -2 \)[/tex] and from [tex]\( -2 \)[/tex] to [tex]\( 0 \)[/tex] and from [tex]\( 0 \)[/tex] to [tex]\( 3.009 \)[/tex].
### Summary
1. Domain:
[tex]\[ [-6, 6] \][/tex]
2. Range:
[tex]\[ (-2, 18.0) \][/tex]
3. Intervals of Increase:
[tex]\[ (-4.006, -2) \][/tex]
[tex]\[ (-2, 0) \][/tex]
[tex]\[ (0, 3.009) \][/tex]
4. Intervals of Decrease:
[tex]\[ (-6, -4.006) \][/tex]
[tex]\[ (3.009, 6) \][/tex]
5. Intervals of Constant Function:
[tex]\[ \text{None} \][/tex]
This step-by-step breakdown outlines the characteristics of the piecewise-defined function.
### Step 1: Define the Function
We have a piecewise function given by:
[tex]\[ f(x) = \begin{cases} 2|x + 4| - 6 & \text{for } -6 \leq x \leq -2 \\ \frac{5}{2} x + 5 & \text{for } -2 < x \leq 0 \\ \frac{-2}{3}|x - 3| + 20 & \text{for } 0 < x \leq 6 \end{cases} \][/tex]
### Step 2: Determine the Domain
The domain of the function is the set of all possible input values [tex]\( x \)[/tex]. According to the piecewise definition, the function is defined from [tex]\( x = -6 \)[/tex] to [tex]\( x = 6 \)[/tex].
Thus, the domain is:
[tex]\[ \text{Domain: } [-6, 6] \][/tex]
### Step 3: Determine the Range
The range of a function is the set of all possible output values [tex]\( f(x) \)[/tex].
From the given answer:
[tex]\[ \text{Range: } (-2, 18.0) \][/tex]
This indicates the lowest possible value of [tex]\( f(x) \)[/tex] is -2 and the highest is 18.0.
### Step 4: Analyze Each Piece of the Function
For [tex]\( -6 \leq x \leq -2 \)[/tex]:
[tex]\[ f(x) = 2|x + 4| - 6 \][/tex]
- [tex]\( |x + 4| \)[/tex] creates a V-shaped graph centered at [tex]\( x = -4 \)[/tex].
- The term [tex]\(-6\)[/tex] shifts this graph downward by 6 units.
- The function decreases from [tex]\( x = -6 \)[/tex] to [tex]\( x = -4 \)[/tex] and then increases from [tex]\( x = -4 \)[/tex] to [tex]\( x = -2 \)[/tex].
For [tex]\( -2 < x \leq 0 \)[/tex]:
[tex]\[ f(x) = \frac{5}{2} x + 5 \][/tex]
- This is a linear function with a slope of [tex]\(\frac{5}{2}\)[/tex] and y-intercept of 5.
- Since the slope is positive, the function is increasing throughout the interval [tex]\( -2 < x \leq 0 \)[/tex].
For [tex]\( 0 < x \leq 6 \)[/tex]:
[tex]\[ f(x) = \frac{-2}{3}|x - 3| + 20 \][/tex]
- [tex]\( |x - 3| \)[/tex] creates a V-shaped graph centered at [tex]\( x = 3 \)[/tex].
- The [tex]\(\frac{-2}{3}\)[/tex] coefficient inverts the V-shape, making it an upside-down V, and the [tex]\( +20 \)[/tex] shifts it upward by 20 units.
- The function increases from [tex]\( x = 0 \)[/tex] to [tex]\( x = 3 \)[/tex] and then decreases from [tex]\( x = 3 \)[/tex] to [tex]\( x = 6 \)[/tex].
### Step 5: Identify Increasing, Constant, or Decreasing Intervals
Based on the given answer:
- Decreasing intervals are from [tex]\( -6 \)[/tex] to [tex]\( -4.006 \)[/tex] and from [tex]\( 3.009 \)[/tex] to [tex]\( 6 \)[/tex].
- Constant intervals do not exist.
- Increasing intervals are from [tex]\( -4.006 \)[/tex] to [tex]\( -2 \)[/tex] and from [tex]\( -2 \)[/tex] to [tex]\( 0 \)[/tex] and from [tex]\( 0 \)[/tex] to [tex]\( 3.009 \)[/tex].
### Summary
1. Domain:
[tex]\[ [-6, 6] \][/tex]
2. Range:
[tex]\[ (-2, 18.0) \][/tex]
3. Intervals of Increase:
[tex]\[ (-4.006, -2) \][/tex]
[tex]\[ (-2, 0) \][/tex]
[tex]\[ (0, 3.009) \][/tex]
4. Intervals of Decrease:
[tex]\[ (-6, -4.006) \][/tex]
[tex]\[ (3.009, 6) \][/tex]
5. Intervals of Constant Function:
[tex]\[ \text{None} \][/tex]
This step-by-step breakdown outlines the characteristics of the piecewise-defined function.
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