Sure, let's analyze each of the given transformations one by one to determine which one matches the transformation of the cube root parent function [tex]\( f(x) = \sqrt[3]{x} \)[/tex].
1. Option A: [tex]\( g(x) = \sqrt[3]{x} + 4 \)[/tex]
- This represents a vertical shift of the parent function upwards by 4 units.
2. Option B: [tex]\( g(x) = \frac{1}{4} \sqrt[3]{x} \)[/tex]
- This represents a vertical compression of the parent function by a factor of [tex]\( \frac{1}{4} \)[/tex].
3. Option C: [tex]\( g(x) = \sqrt[3]{x + 4} \)[/tex]
- This represents a horizontal shift of the parent function to the left by 4 units. For horizontal transformations, the shift to the left is achieved by adding a constant inside the function.
4. Option D: [tex]\( g(x) = 4 \sqrt[3]{x} \)[/tex]
- This represents a vertical stretch of the parent function by a factor of 4.
Based on these interpretations of each transformation:
- A vertical shift would change our function to [tex]\( \sqrt[3]{x} + 4 \)[/tex].
- A vertical compression would change our function to [tex]\( \frac{1}{4} \sqrt[3]{x} \)[/tex].
- A horizontal shift left would change our function to [tex]\( \sqrt[3]{x + 4} \)[/tex].
- A vertical stretch would change our function to [tex]\( 4 \sqrt[3]{x} \)[/tex].
The correct transformation we are looking for is a shift that modifies the input of the parent function [tex]\( f(x) \)[/tex]. The correct answer is when the function inside the cube root has been adjusted to [tex]\( x + 4 \)[/tex], indicating a horizontal shift left by 4 units.
Thus, the correct transformed function [tex]\( g(x) \)[/tex] is:
[tex]\[ \boxed{g(x) = \sqrt[3]{x + 4}} \][/tex]
This matches with Option C.
