To solve this problem, let's analyze the function transformation given:
[tex]\[ g(x) = f\left(\frac{1}{3} x\right) \][/tex]
This type of transformation involves the horizontal scaling of the graph of the function [tex]\( f(x) \)[/tex].
1. Understanding Function Transformations:
When we see a function in the form [tex]\( g(x) = f(bx) \)[/tex], it tells us that there is a horizontal scaling transformation. To determine the scale factor, we need to consider the value of [tex]\( b \)[/tex]:
- If [tex]\( 0 < b < 1 \)[/tex]: The graph of [tex]\( f(x) \)[/tex] is stretched horizontally by a scale factor of [tex]\( \frac{1}{b} \)[/tex].
- If [tex]\( b > 1 \)[/tex]: The graph of [tex]\( f(x) \)[/tex] is compressed horizontally by a scale factor of [tex]\( \frac{1}{b} \)[/tex].
2. Applying This Concept to Our Problem:
In our case, the transformation is [tex]\( g(x) = f\left(\frac{1}{3} x\right) \)[/tex], which gives us [tex]\( b = \frac{1}{3} \)[/tex].
- Since [tex]\( b = \frac{1}{3} \)[/tex] and [tex]\( 0 < \frac{1}{3} < 1 \)[/tex], this indicates that the graph of [tex]\( f(x) \)[/tex] will be stretched horizontally.
- The stretch factor is [tex]\( \frac{1}{b} \)[/tex]. So, we calculate [tex]\( \frac{1}{\frac{1}{3}} = 3 \)[/tex].
Thus, 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]. This corresponds to option B.
So, the correct answer is:
[tex]\[ \boxed{B} \][/tex]
