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If [tex]g(x)=f(4x)[/tex], which statement is true?

A. The graph of function [tex]f[/tex] is stretched horizontally by a scale factor of 4 to create the graph of function [tex]g[/tex].

B. The graph of function [tex]f[/tex] is compressed vertically by a scale factor of [tex]\frac{1}{4}[/tex] to create the graph of function [tex]g[/tex].

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].

D. The graph of function [tex]f[/tex] is stretched vertically by a scale factor of 4 to create the graph of function [tex]g[/tex].

Sagot :

GinnyAnswer
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.
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