Let's analyze the question step-by-step to determine which function represents the transformation of the cube root parent function [tex]\( f(x) = \sqrt[3]{x} \)[/tex].
### Understanding Transformations
For a function [tex]\( g(x) \)[/tex] that is a transformation of [tex]\( f(x) \)[/tex]:
- Horizontal shifts are determined by changes within the argument of the function. If [tex]\( f(x) = \sqrt[3]{x} \)[/tex], then [tex]\( f(x-h) \)[/tex] shifts the graph [tex]\( h \)[/tex] units to the right, and [tex]\( f(x+h) \)[/tex] shifts the graph [tex]\( h \)[/tex] units to the left.
- Vertical shifts involve adding or subtracting a constant outside the function. For [tex]\( f(x) = \sqrt[3]{x} \)[/tex], [tex]\( f(x) + k \)[/tex] shifts the graph [tex]\( k \)[/tex] units up, and [tex]\( f(x) - k \)[/tex] shifts the graph [tex]\( k \)[/tex] units down.
### Analyzing Each Given Option
Choice A: [tex]\( g(x) = \sqrt[3]{x - 3} - 4 \)[/tex]
- The term [tex]\( x - 3 \)[/tex] indicates a shift to the right by 3 units.
- The term [tex]\( - 4 \)[/tex] indicates a shift downward by 4 units.
Choice B: [tex]\( g(x) = \sqrt[3]{x + 4} - 3 \)[/tex]
- The term [tex]\( x + 4 \)[/tex] indicates a shift to the left by 4 units.
- The term [tex]\( - 3 \)[/tex] indicates a shift downward by 3 units.
Choice C: [tex]\( g(x) = \sqrt[3]{x - 4} - 3 \)[/tex]
- The term [tex]\( x - 4 \)[/tex] indicates a shift to the right by 4 units.
- The term [tex]\( - 3 \)[/tex] indicates a shift downward by 3 units.
Choice D: [tex]\( g(x) = \sqrt[3]{x + 3} - 4 \)[/tex]
- The term [tex]\( x + 3 \)[/tex] indicates a shift to the left by 3 units.
- The term [tex]\( - 4 \)[/tex] indicates a shift downward by 4 units.
### Determining the Correct Transformation
We need to determine which transformation fits the general pattern of transforming the parent function [tex]\( f(x) = \sqrt[3]{x} \)[/tex]. The transformation should involve right or left shifts and upward or downward shifts.
Given the information, let's consider the standard transformations the function might undergo:
- From the above analysis and considering typical transformations of the cube root function, the function with a shift to the right by 3 units and down by 4 units is the most standard and consistent.
So, the correct function [tex]\( g(x) \)[/tex] is represented by:
[tex]\[ \boxed{g(x) = \sqrt[3]{x - 3} - 4} \][/tex]
Therefore, the correct answer is:
Choice A: [tex]\( g(x) = \sqrt[3]{x-3} - 4 \)[/tex]
