Answer :
To determine how the function [tex]\( f(x) = \sqrt[5]{x} \)[/tex] is transformed into [tex]\( g(x) \)[/tex] given the options, we need to analyze the transformations in the provided form: [tex]\( g(x) = a \cdot f(x+b) \)[/tex].
### Step-by-step Analysis:
1. Identify Vertical Scaling and Reflection:
- The coefficient [tex]\( a \)[/tex] in front of the function [tex]\( f \)[/tex] affects the vertical scaling and reflection of the graph. When [tex]\( a \)[/tex] is negative, the graph reflects over the x-axis.
- If [tex]\( a \)[/tex] is -2, the transformation involves reflecting the graph over the x-axis and scaling it vertically by a factor of 2. This combines the effect of reflection and stretching by 2.
2. Identify Horizontal Translation:
- The term [tex]\( x+b \)[/tex] inside the function shifts the graph horizontally.
- If [tex]\( b \)[/tex] is negative, the graph shifts to the left by [tex]\(|b|\)[/tex] units.
- If [tex]\( b \)[/tex] is positive, the graph shifts to the right by [tex]\(|b|\)[/tex] units.
- Therefore, in the expression [tex]\( g(x) = a \cdot f(x+b) \)[/tex], if [tex]\( x+b \)[/tex] is [tex]\( x+4 \)[/tex], then the graph of [tex]\( f(x) \)[/tex] shifts to the left by 4 units.
### Matching the Analysis to the Given Options:
Given the options are:
- [tex]\( -2 f(x+4) \)[/tex]
- [tex]\( -2 f(x-4) \)[/tex]
- [tex]\( -f(x+4) \)[/tex]
- [tex]\( -f(x-4) \)[/tex]
Let's analyze each option:
1. Option 1: [tex]\( g(x) = -2 f(x+4) \)[/tex]
- [tex]\( a = -2 \)[/tex]: Vertically scales by 2 and reflects over the x-axis.
- [tex]\( b = 4 \)[/tex]: Shifts the graph 4 units to the left.
- Hence, this represents the transformation involving both vertical scaling by -2 and horizontal translation to the left by 4 units.
2. Option 2: [tex]\( g(x) = -2 f(x-4) \)[/tex]
- [tex]\( a = -2 \)[/tex]: Vertically scales by 2 and reflects over the x-axis.
- [tex]\( b = -4 \)[/tex]: Shifts the graph 4 units to the right.
- This transformation is not consistent with our requirement of shifting 4 units to the left.
3. Option 3: [tex]\( g(x) = -f(x+4) \)[/tex]
- [tex]\( a = -1 \)[/tex]: Reflects over the x-axis without any vertical scaling.
- [tex]\( b = 4 \)[/tex]: Shifts the graph 4 units to the left.
- This transformation doesn't include scaling by 2.
4. Option 4: [tex]\( g(x) = -f(x-4) \)[/tex]
- [tex]\( a = -1 \)[/tex]: Reflects over the x-axis without any vertical scaling.
- [tex]\( b = -4 \)[/tex]: Shifts the graph 4 units to the right.
- This transformation is also incorrect due to direction of the shift and lack of vertical scaling by 2.
### Conclusion:
The correct transformation of [tex]\( f(x) = \sqrt[5]{x} \)[/tex] to get [tex]\( g(x) \)[/tex] is given by:
[tex]\[ g(x) = -2 f(x+4) \][/tex]
So the right answer is the first option:
- [tex]\( -2 f(x + 4) \)[/tex].
### Step-by-step Analysis:
1. Identify Vertical Scaling and Reflection:
- The coefficient [tex]\( a \)[/tex] in front of the function [tex]\( f \)[/tex] affects the vertical scaling and reflection of the graph. When [tex]\( a \)[/tex] is negative, the graph reflects over the x-axis.
- If [tex]\( a \)[/tex] is -2, the transformation involves reflecting the graph over the x-axis and scaling it vertically by a factor of 2. This combines the effect of reflection and stretching by 2.
2. Identify Horizontal Translation:
- The term [tex]\( x+b \)[/tex] inside the function shifts the graph horizontally.
- If [tex]\( b \)[/tex] is negative, the graph shifts to the left by [tex]\(|b|\)[/tex] units.
- If [tex]\( b \)[/tex] is positive, the graph shifts to the right by [tex]\(|b|\)[/tex] units.
- Therefore, in the expression [tex]\( g(x) = a \cdot f(x+b) \)[/tex], if [tex]\( x+b \)[/tex] is [tex]\( x+4 \)[/tex], then the graph of [tex]\( f(x) \)[/tex] shifts to the left by 4 units.
### Matching the Analysis to the Given Options:
Given the options are:
- [tex]\( -2 f(x+4) \)[/tex]
- [tex]\( -2 f(x-4) \)[/tex]
- [tex]\( -f(x+4) \)[/tex]
- [tex]\( -f(x-4) \)[/tex]
Let's analyze each option:
1. Option 1: [tex]\( g(x) = -2 f(x+4) \)[/tex]
- [tex]\( a = -2 \)[/tex]: Vertically scales by 2 and reflects over the x-axis.
- [tex]\( b = 4 \)[/tex]: Shifts the graph 4 units to the left.
- Hence, this represents the transformation involving both vertical scaling by -2 and horizontal translation to the left by 4 units.
2. Option 2: [tex]\( g(x) = -2 f(x-4) \)[/tex]
- [tex]\( a = -2 \)[/tex]: Vertically scales by 2 and reflects over the x-axis.
- [tex]\( b = -4 \)[/tex]: Shifts the graph 4 units to the right.
- This transformation is not consistent with our requirement of shifting 4 units to the left.
3. Option 3: [tex]\( g(x) = -f(x+4) \)[/tex]
- [tex]\( a = -1 \)[/tex]: Reflects over the x-axis without any vertical scaling.
- [tex]\( b = 4 \)[/tex]: Shifts the graph 4 units to the left.
- This transformation doesn't include scaling by 2.
4. Option 4: [tex]\( g(x) = -f(x-4) \)[/tex]
- [tex]\( a = -1 \)[/tex]: Reflects over the x-axis without any vertical scaling.
- [tex]\( b = -4 \)[/tex]: Shifts the graph 4 units to the right.
- This transformation is also incorrect due to direction of the shift and lack of vertical scaling by 2.
### Conclusion:
The correct transformation of [tex]\( f(x) = \sqrt[5]{x} \)[/tex] to get [tex]\( g(x) \)[/tex] is given by:
[tex]\[ g(x) = -2 f(x+4) \][/tex]
So the right answer is the first option:
- [tex]\( -2 f(x + 4) \)[/tex].
