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Certainly! Let's break down the piece-wise function [tex]\( f(x) \)[/tex] into its separate cases and then graph it. We will also determine the domain and range step-by-step.
### Piece-wise Function Definition
The function [tex]\( f(x) \)[/tex] is defined as:
[tex]\[ f(x) = \begin{cases} \sqrt{x - 2} & \text{for } x < -2 \\ x^2 & \text{for } -2 \leq x \leq 2 \\ -4 & \text{for } x > 2 \end{cases} \][/tex]
### Step-by-Step Analysis
1. Case 1: [tex]\( x < -2 \)[/tex]
Here, [tex]\( f(x) = \sqrt{x - 2} \)[/tex].
Note that the expression [tex]\( \sqrt{x - 2} \)[/tex] is not valid for [tex]\( x < -2 \)[/tex] since the square root of a negative number is not defined in the real number system. Therefore, we can infer this part of the function is actually not applicable in the real number domain.
2. Case 2: [tex]\( -2 \leq x \leq 2 \)[/tex]
Here, [tex]\( f(x) = x^2 \)[/tex].
- At [tex]\( x = -2 \)[/tex]: [tex]\( f(-2) = (-2)^2 = 4 \)[/tex]
- At [tex]\( x = 2 \)[/tex]: [tex]\( f(2) = 2^2 = 4 \)[/tex]
- For all [tex]\( x \)[/tex] in the range [tex]\( -2 \leq x \leq 2 \)[/tex], [tex]\( f(x) \)[/tex] will be the square of [tex]\( x \)[/tex]. This forms a parabola opening upwards from [tex]\( x = -2 \)[/tex] to [tex]\( x = 2 \)[/tex].
3. Case 3: [tex]\( x > 2 \)[/tex]
Here, [tex]\( f(x) = -4 \)[/tex].
For any [tex]\( x > 2 \)[/tex], [tex]\( f(x) = -4 \)[/tex], which is a constant function.
### Domain and Range
- Domain: The domain of [tex]\( f(x) \)[/tex] includes all [tex]\( x \)[/tex] values for which [tex]\( f(x) \)[/tex] is defined. From our analysis:
- [tex]\( \sqrt{x - 2} \)[/tex] is not defined for [tex]\( x < -2 \)[/tex].
- [tex]\( x^2 \)[/tex] is defined for [tex]\( -2 \leq x \leq 2 \)[/tex].
- [tex]\(-4\)[/tex] is defined for [tex]\( x > 2 \)[/tex].
Thus, the domain is:
[tex]\[ \text{Dom}(f) = [-2, \infty) \][/tex]
- Range: The range of [tex]\( f(x) \)[/tex] includes all [tex]\( y \)[/tex] values that [tex]\( f(x) \)[/tex] can take:
- For [tex]\( -2 \leq x \leq 2 \)[/tex]: Since [tex]\( f(x) = x^2 \)[/tex], the output ranges from 0 to 4. This forms the interval [tex]\([0, 4]\)[/tex].
- For [tex]\( x > 2 \)[/tex]: [tex]\( f(x) = -4 \)[/tex], so the value -4 is also included in the range.
Thus, the range is:
[tex]\[ \text{Range}(f) = [-4, 4] \][/tex]
### Graph of [tex]\( f(x) \)[/tex]
To graph this function:
1. For [tex]\( -2 \leq x \leq 2 \)[/tex]:
- Plot the parabola [tex]\( y = x^2 \)[/tex] from [tex]\( x = -2 \)[/tex] to [tex]\( x = 2 \)[/tex].
2. For [tex]\( x > 2 \)[/tex]:
- Plot the constant line [tex]\( y = -4 \)[/tex] for [tex]\( x > 2 \)[/tex].
### Summary
- Domain: [tex]\([-2, \infty)\)[/tex]
- Range: [tex]\([-4, 4]\)[/tex]
### The Graph
The graph of the given piece-wise function will look like:
- A parabola from [tex]\((-2, 4)\)[/tex] to [tex]\((2, 4)\)[/tex].
- A horizontal line at [tex]\(y = -4\)[/tex] starting from [tex]\(x = 2\)[/tex].
This is a simplified explanation without the visualization, but this step-by-step approach should help in understanding how to graph the piece-wise function and determine its domain and range.
### Piece-wise Function Definition
The function [tex]\( f(x) \)[/tex] is defined as:
[tex]\[ f(x) = \begin{cases} \sqrt{x - 2} & \text{for } x < -2 \\ x^2 & \text{for } -2 \leq x \leq 2 \\ -4 & \text{for } x > 2 \end{cases} \][/tex]
### Step-by-Step Analysis
1. Case 1: [tex]\( x < -2 \)[/tex]
Here, [tex]\( f(x) = \sqrt{x - 2} \)[/tex].
Note that the expression [tex]\( \sqrt{x - 2} \)[/tex] is not valid for [tex]\( x < -2 \)[/tex] since the square root of a negative number is not defined in the real number system. Therefore, we can infer this part of the function is actually not applicable in the real number domain.
2. Case 2: [tex]\( -2 \leq x \leq 2 \)[/tex]
Here, [tex]\( f(x) = x^2 \)[/tex].
- At [tex]\( x = -2 \)[/tex]: [tex]\( f(-2) = (-2)^2 = 4 \)[/tex]
- At [tex]\( x = 2 \)[/tex]: [tex]\( f(2) = 2^2 = 4 \)[/tex]
- For all [tex]\( x \)[/tex] in the range [tex]\( -2 \leq x \leq 2 \)[/tex], [tex]\( f(x) \)[/tex] will be the square of [tex]\( x \)[/tex]. This forms a parabola opening upwards from [tex]\( x = -2 \)[/tex] to [tex]\( x = 2 \)[/tex].
3. Case 3: [tex]\( x > 2 \)[/tex]
Here, [tex]\( f(x) = -4 \)[/tex].
For any [tex]\( x > 2 \)[/tex], [tex]\( f(x) = -4 \)[/tex], which is a constant function.
### Domain and Range
- Domain: The domain of [tex]\( f(x) \)[/tex] includes all [tex]\( x \)[/tex] values for which [tex]\( f(x) \)[/tex] is defined. From our analysis:
- [tex]\( \sqrt{x - 2} \)[/tex] is not defined for [tex]\( x < -2 \)[/tex].
- [tex]\( x^2 \)[/tex] is defined for [tex]\( -2 \leq x \leq 2 \)[/tex].
- [tex]\(-4\)[/tex] is defined for [tex]\( x > 2 \)[/tex].
Thus, the domain is:
[tex]\[ \text{Dom}(f) = [-2, \infty) \][/tex]
- Range: The range of [tex]\( f(x) \)[/tex] includes all [tex]\( y \)[/tex] values that [tex]\( f(x) \)[/tex] can take:
- For [tex]\( -2 \leq x \leq 2 \)[/tex]: Since [tex]\( f(x) = x^2 \)[/tex], the output ranges from 0 to 4. This forms the interval [tex]\([0, 4]\)[/tex].
- For [tex]\( x > 2 \)[/tex]: [tex]\( f(x) = -4 \)[/tex], so the value -4 is also included in the range.
Thus, the range is:
[tex]\[ \text{Range}(f) = [-4, 4] \][/tex]
### Graph of [tex]\( f(x) \)[/tex]
To graph this function:
1. For [tex]\( -2 \leq x \leq 2 \)[/tex]:
- Plot the parabola [tex]\( y = x^2 \)[/tex] from [tex]\( x = -2 \)[/tex] to [tex]\( x = 2 \)[/tex].
2. For [tex]\( x > 2 \)[/tex]:
- Plot the constant line [tex]\( y = -4 \)[/tex] for [tex]\( x > 2 \)[/tex].
### Summary
- Domain: [tex]\([-2, \infty)\)[/tex]
- Range: [tex]\([-4, 4]\)[/tex]
### The Graph
The graph of the given piece-wise function will look like:
- A parabola from [tex]\((-2, 4)\)[/tex] to [tex]\((2, 4)\)[/tex].
- A horizontal line at [tex]\(y = -4\)[/tex] starting from [tex]\(x = 2\)[/tex].
This is a simplified explanation without the visualization, but this step-by-step approach should help in understanding how to graph the piece-wise function and determine its domain and range.
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