Answer :
To solve the problem of finding the values of the function [tex]\( f(x) \)[/tex] for specific [tex]\( x \)[/tex] values, let's proceed step-by-step.
We are given a piecewise function [tex]\( f(x) \)[/tex] defined as follows:
[tex]\[ f(x) = \begin{cases} \sqrt{-4x}, & \text{if } x \neq -4 \\ -6, & \text{if } x = -4 \end{cases} \][/tex]
We need to evaluate this function for a set of given [tex]\( x \)[/tex] values: [tex]\(-4, -2, 0, 2, 4\)[/tex].
1. When [tex]\( x = -4 \)[/tex]:
- According to the definition of the function, when [tex]\( x = -4 \)[/tex], [tex]\( f(x) = -6 \)[/tex].
- Therefore, [tex]\( f(-4) = -6 \)[/tex].
2. When [tex]\( x = -2 \)[/tex]:
- Since [tex]\( x \neq -4 \)[/tex], we use the first case of the function definition.
- We have [tex]\( f(x) = \sqrt{-4x} \)[/tex]. Substituting [tex]\( x = -2 \)[/tex], we get:
[tex]\( f(-2) = \sqrt{-4 \cdot (-2)} = \sqrt{8} \approx 2.8284271247461903 \)[/tex].
- Therefore, [tex]\( f(-2) \approx 2.8284271247461903 \)[/tex].
3. When [tex]\( x = 0 \)[/tex]:
- Since [tex]\( x \neq -4 \)[/tex], we use the first case of the function definition.
- We have [tex]\( f(x) = \sqrt{-4x} \)[/tex]. Substituting [tex]\( x = 0 \)[/tex], we get:
[tex]\( f(0) = \sqrt{-4 \cdot 0} = \sqrt{0} = 0.0 \)[/tex].
- Therefore, [tex]\( f(0) = 0.0 \)[/tex].
4. When [tex]\( x = 2 \)[/tex]:
- Since [tex]\( x \neq -4 \)[/tex], we use the first case of the function definition.
- We have [tex]\( f(x) = \sqrt{-4x} \)[/tex]. Substituting [tex]\( x = 2 \)[/tex], we get:
[tex]\( f(2) = \sqrt{-4 \cdot 2} = \sqrt{-8} \)[/tex].
- [tex]\(\sqrt{-8}\)[/tex] is an imaginary number because we are taking the square root of a negative number.
- Specifically, [tex]\( \sqrt{-8} = \sqrt{-1 \cdot 8} = \sqrt{8} \cdot i \)[/tex].
- This simplifies to approximately [tex]\( 2.8284271247461903i \)[/tex].
- Therefore, [tex]\( f(2) \approx 1.7319121124709868e-16 + 2.8284271247461903i \)[/tex].
5. When [tex]\( x = 4 \)[/tex]:
- Since [tex]\( x \neq -4 \)[/tex], we use the first case of the function definition.
- We have [tex]\( f(x) = \sqrt{-4x} \)[/tex]. Substituting [tex]\( x = 4 \)[/tex], we get:
[tex]\( f(4) = \sqrt{-4 \cdot 4} = \sqrt{-16} \)[/tex].
- [tex]\(\sqrt{-16}\)[/tex] is an imaginary number because we are taking the square root of a negative number.
- Specifically, [tex]\( \sqrt{-16} = \sqrt{-1 \cdot 16} = \sqrt{16} \cdot i \)[/tex].
- This simplifies to [tex]\( 4i \)[/tex].
- Therefore, [tex]\( f(4) \approx 2.4492935982947064e-16 + 4i \)[/tex].
Summary of the results:
[tex]\[ \begin{aligned} &f(-4) = -6, \\ &f(-2) \approx 2.8284271247461903, \\ &f(0) = 0.0, \\ &f(2) \approx 1.7319121124709868e-16 + 2.8284271247461903i, \\ &f(4) \approx 2.4492935982947064e-16 + 4i. \end{aligned} \][/tex]
These are the function values corresponding to the input values [tex]\(-4, -2, 0, 2, 4\)[/tex].
We are given a piecewise function [tex]\( f(x) \)[/tex] defined as follows:
[tex]\[ f(x) = \begin{cases} \sqrt{-4x}, & \text{if } x \neq -4 \\ -6, & \text{if } x = -4 \end{cases} \][/tex]
We need to evaluate this function for a set of given [tex]\( x \)[/tex] values: [tex]\(-4, -2, 0, 2, 4\)[/tex].
1. When [tex]\( x = -4 \)[/tex]:
- According to the definition of the function, when [tex]\( x = -4 \)[/tex], [tex]\( f(x) = -6 \)[/tex].
- Therefore, [tex]\( f(-4) = -6 \)[/tex].
2. When [tex]\( x = -2 \)[/tex]:
- Since [tex]\( x \neq -4 \)[/tex], we use the first case of the function definition.
- We have [tex]\( f(x) = \sqrt{-4x} \)[/tex]. Substituting [tex]\( x = -2 \)[/tex], we get:
[tex]\( f(-2) = \sqrt{-4 \cdot (-2)} = \sqrt{8} \approx 2.8284271247461903 \)[/tex].
- Therefore, [tex]\( f(-2) \approx 2.8284271247461903 \)[/tex].
3. When [tex]\( x = 0 \)[/tex]:
- Since [tex]\( x \neq -4 \)[/tex], we use the first case of the function definition.
- We have [tex]\( f(x) = \sqrt{-4x} \)[/tex]. Substituting [tex]\( x = 0 \)[/tex], we get:
[tex]\( f(0) = \sqrt{-4 \cdot 0} = \sqrt{0} = 0.0 \)[/tex].
- Therefore, [tex]\( f(0) = 0.0 \)[/tex].
4. When [tex]\( x = 2 \)[/tex]:
- Since [tex]\( x \neq -4 \)[/tex], we use the first case of the function definition.
- We have [tex]\( f(x) = \sqrt{-4x} \)[/tex]. Substituting [tex]\( x = 2 \)[/tex], we get:
[tex]\( f(2) = \sqrt{-4 \cdot 2} = \sqrt{-8} \)[/tex].
- [tex]\(\sqrt{-8}\)[/tex] is an imaginary number because we are taking the square root of a negative number.
- Specifically, [tex]\( \sqrt{-8} = \sqrt{-1 \cdot 8} = \sqrt{8} \cdot i \)[/tex].
- This simplifies to approximately [tex]\( 2.8284271247461903i \)[/tex].
- Therefore, [tex]\( f(2) \approx 1.7319121124709868e-16 + 2.8284271247461903i \)[/tex].
5. When [tex]\( x = 4 \)[/tex]:
- Since [tex]\( x \neq -4 \)[/tex], we use the first case of the function definition.
- We have [tex]\( f(x) = \sqrt{-4x} \)[/tex]. Substituting [tex]\( x = 4 \)[/tex], we get:
[tex]\( f(4) = \sqrt{-4 \cdot 4} = \sqrt{-16} \)[/tex].
- [tex]\(\sqrt{-16}\)[/tex] is an imaginary number because we are taking the square root of a negative number.
- Specifically, [tex]\( \sqrt{-16} = \sqrt{-1 \cdot 16} = \sqrt{16} \cdot i \)[/tex].
- This simplifies to [tex]\( 4i \)[/tex].
- Therefore, [tex]\( f(4) \approx 2.4492935982947064e-16 + 4i \)[/tex].
Summary of the results:
[tex]\[ \begin{aligned} &f(-4) = -6, \\ &f(-2) \approx 2.8284271247461903, \\ &f(0) = 0.0, \\ &f(2) \approx 1.7319121124709868e-16 + 2.8284271247461903i, \\ &f(4) \approx 2.4492935982947064e-16 + 4i. \end{aligned} \][/tex]
These are the function values corresponding to the input values [tex]\(-4, -2, 0, 2, 4\)[/tex].
