To solve the problem, we need to understand the transformation of the function [tex]\( f(x) \)[/tex] to [tex]\( g(x) = F(4x) \)[/tex].
Let's break down what happens when a function's input is scaled.
1. Horizontal Scaling:
- When [tex]\( g(x) = f(ax) \)[/tex], the function experiences a horizontal transformation.
- If [tex]\( a > 1 \)[/tex], the graph of the function is compressed horizontally by a factor of [tex]\( \frac{1}{a} \)[/tex].
- If [tex]\( 0 < a < 1 \)[/tex], the graph of the function is stretched horizontally by a factor of [tex]\( \frac{1}{a} \)[/tex].
In this case, [tex]\( g(x) = f(4x) \)[/tex]:
- Here, [tex]\( a = 4 \)[/tex].
Since [tex]\( a = 4 \)[/tex] is greater than 1, the graph of [tex]\( g(x) \)[/tex] is a horizontal transformation of [tex]\( f(x) \)[/tex] that compresses it by a factor of [tex]\( \frac{1}{4} \)[/tex].
Therefore, the correct statement is:
C. The graph of function [tex]\( f \)[/tex] is compressed horizontally by a scale factor of [tex]\( \frac{1}{4} \)[/tex] to create the graph of function [tex]\( g \)[/tex].
So, the correct answer is C.
