To determine the correct transformation of the function [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex] where [tex]\( g(x) = f\left(\frac{1}{3} x\right) \)[/tex], let's analyze how the given transformation affects the graph:
1. Understand the Transformation:
- The function [tex]\( g(x) \)[/tex] is defined as [tex]\( f\left(\frac{1}{3} x\right) \)[/tex].
- Here, [tex]\( \frac{1}{3} x \)[/tex] indicates that the input [tex]\( x \)[/tex] in the function [tex]\( f \)[/tex] is being multiplied by [tex]\( \frac{1}{3} \)[/tex].
2. Effect on the Graph:
- When the input [tex]\( x \)[/tex] is multiplied by a number [tex]\( k \)[/tex] inside the function [tex]\( f \)[/tex], the graph of [tex]\( f(x) \)[/tex] is affected horizontally.
- Specifically, if [tex]\( k = \frac{1}{3} \)[/tex], this implies a horizontal transformation.
3. Horizontal Scaling:
- If [tex]\( k \)[/tex] is a fraction between 0 and 1, it typically compresses the graph horizontally. However, because the multiplication here is inside the function (i.e., [tex]\( f \)[/tex] is applied to [tex]\( \frac{1}{3} x \)[/tex]), it results in expanding the graph of [tex]\( f(x) \)[/tex] by a factor of [tex]\( \frac{1}{k} \)[/tex], which in this case is 3.
- Therefore, the graph of the function [tex]\( f \)[/tex] is stretched horizontally by a factor of 3.
4. Conclusion:
- Among the given choices, the correct statement describing this transformation is:
B. The graph of function [tex]\( f \)[/tex] is stretched horizontally by a scale factor of 3 to create the graph of function [tex]\( g \)[/tex].
So, the correct answer is B.
