Let's start by understanding how each transformation affects the cube root parent function, [tex]\( f(x) = \sqrt[3]{x} \)[/tex].
1. Horizontal Shifts:
- Replacing [tex]\( x \)[/tex] with [tex]\( x - h \)[/tex] shifts the function to the right by [tex]\( h \)[/tex] units.
- Replacing [tex]\( x \)[/tex] with [tex]\( x + h \)[/tex] shifts the function to the left by [tex]\( h \)[/tex] units.
2. Vertical Shifts:
- Adding [tex]\( k \)[/tex] to the function, [tex]\( \sqrt[3]{x} + k \)[/tex], shifts the function up by [tex]\( k \)[/tex] units.
- Subtracting [tex]\( k \)[/tex] from the function, [tex]\( \sqrt[3]{x} - k \)[/tex], shifts the function down by [tex]\( k \)[/tex] units.
Now let's analyze each given option:
A. [tex]\( g(x) = \sqrt[3]{x - 3} + 4 \)[/tex]
- This transformation shifts the parent function [tex]\( \sqrt[3]{x} \)[/tex] to the right by 3 units and up by 4 units.
B. [tex]\( g(x) = \sqrt[3]{x + 3} + 4 \)[/tex]
- This transformation shifts the parent function [tex]\( \sqrt[3]{x} \)[/tex] to the left by 3 units and up by 4 units.
C. [tex]\( g(x) = \sqrt[3]{x + 4} + 3 \)[/tex]
- This transformation shifts the parent function [tex]\( \sqrt[3]{x} \)[/tex] to the left by 4 units and up by 3 units.
D. [tex]\( g(x) = \sqrt[3]{x - 4} + 3 \)[/tex]
- This transformation shifts the parent function [tex]\( \sqrt[3]{x} \)[/tex] to the right by 4 units and up by 3 units.
Given this information, we need to determine the correct transformation among these options.
From the analysis:
- [tex]\( g(x) = \sqrt[3]{x - 3} + 4 \)[/tex], option A, correctly describes a shift to the right by 3 units and a vertical shift upwards by 4 units.
Thus, the appropriate function [tex]\( g(x) \)[/tex] is:
[tex]\[ \boxed{g(x) = \sqrt[3]{x-3} + 4} \][/tex]
