The function [tex]\( g(x) \)[/tex] is a transformation of the cube root parent function, [tex]\( f(x) = \sqrt[3]{x} \)[/tex].
We have two possible options:
A. [tex]\( g(x) = 4 \sqrt[3]{x} \)[/tex]
B. [tex]\( g(x) = \sqrt[3]{x} + 4 \)[/tex]
Let's analyze each option to find the correct transformation:
1. Option A: [tex]\( g(x) = 4 \sqrt[3]{x} \)[/tex]
In this case, [tex]\( g(x) \)[/tex] is obtained by multiplying the parent function [tex]\( f(x) = \sqrt[3]{x} \)[/tex] by 4. This kind of transformation is called a vertical stretch. When you multiply a function by a constant, you are stretching or compressing its graph vertically by that constant factor. Here, the graph of [tex]\( f(x) \)[/tex] is stretched vertically by a factor of 4.
2. Option B: [tex]\( g(x) = \sqrt[3]{x} + 4 \)[/tex]
In this case, [tex]\( g(x) \)[/tex] is obtained by adding 4 to the parent function [tex]\( f(x) = \sqrt[3]{x} \)[/tex]. This kind of transformation is called a vertical translation. When you add a constant to a function, you are shifting its graph up or down by that constant amount. Here, the graph of [tex]\( f(x) \)[/tex] is translated upwards by 4 units.
Since we need to determine the transformation that describes [tex]\( g(x) \)[/tex] correctly, let's summarize:
- Option A represents a vertical stretch of the parent function by a factor of 4.
- Option B represents a vertical translation of the parent function up by 4 units.
Given these characteristics of transformation, the correct function [tex]\( g(x) \)[/tex] that matches the description is:
[tex]\[ g(x) = 4 \sqrt[3]{x} \][/tex]
So, the correct option is:
A. [tex]\( g(x) = 4 \sqrt[3]{x} \)[/tex]
