To determine which of the given functions, [tex]\( g(x) \)[/tex], is a transformation of the cube root parent function [tex]\( f(x) = \sqrt[3]{x} \)[/tex], let's analyze each choice step-by-step in detail to find the correct transformation.
### Analysis of Each Choice:
1. Choice A: [tex]\( g(x) = 4\sqrt[3]{x} \)[/tex]
- This function involves multiplying the cube root function by 4.
- This transformation results in a vertical stretch of the parent function [tex]\( f(x) \)[/tex] by a factor of 4.
- The graph of [tex]\( g(x) = 4\sqrt[3]{x} \)[/tex] is stretched vertically compared to [tex]\( f(x) \)[/tex].
2. Choice B: [tex]\( g(x) = \sqrt[3]{x} + 4 \)[/tex]
- This function involves adding 4 to the cube root function.
- This transformation shifts the entire graph of the parent function [tex]\( f(x) \)[/tex] upward by 4 units.
- The graph of [tex]\( g(x) = \sqrt[3]{x} + 4 \)[/tex] moves up compared to [tex]\( f(x) \)[/tex].
3. Choice C: [tex]\( g(x) = \sqrt[3]{x + 4} \)[/tex]
- This function involves adding 4 to the input [tex]\( x \)[/tex] before taking the cube root.
- This transformation shifts the graph of the parent function [tex]\( f(x) \)[/tex] to the left by 4 units.
- The graph of [tex]\( g(x) = \sqrt[3]{x+4} \)[/tex] moves to the left compared to [tex]\( f(x) \)[/tex].
4. Choice D: [tex]\( g(x) = \frac{1}{4}\sqrt[3]{x} \)[/tex]
- This function involves multiplying the cube root function by [tex]\( \frac{1}{4} \)[/tex].
- This transformation results in a vertical compression of the parent function [tex]\( f(x) \)[/tex] by a factor of 1/4.
- The graph of [tex]\( g(x) = \frac{1}{4}\sqrt[3]{x} \)[/tex] is compressed vertically compared to [tex]\( f(x) \)[/tex].
### Conclusion:
By examining the different transformations, the function [tex]\( g(x) = \sqrt[3]{x+4} \)[/tex] (Choice C) involves shifting the graph to the left by 4 units, which matches our given objective.
Therefore, the function [tex]\( g(x) \)[/tex] as a transformation of the cube root parent function [tex]\( f(x) = \sqrt[3]{x} \)[/tex] is:
[tex]\[ g(x) = \sqrt[3]{x+4} \][/tex]
