Let's look at the given problem statement:
We are given the cube root parent function [tex]\( f(x) = \sqrt[3]{x} \)[/tex], and we need to determine the function [tex]\( g(x) \)[/tex] that results from specific transformations of this parent function.
We need [tex]\( g(x) = \sqrt[3]{x-2} + 1 \)[/tex].
Let's break down the transformation step by step:
1. Horizontal Shift by 2 Units to the Right:
- Instead of [tex]\( f(x) \)[/tex], consider [tex]\( f(x-2) \)[/tex].
- This operation shifts the graph of [tex]\( \sqrt[3]{x} \)[/tex] 2 units to the right.
- Therefore, [tex]\( f(x-2) = \sqrt[3]{x-2} \)[/tex].
2. Vertical Shift by 1 Unit Upward:
- Now, we take [tex]\( f(x-2) = \sqrt[3]{x-2} \)[/tex] and add 1.
- This operation shifts the graph of [tex]\( \sqrt[3]{x-2} \)[/tex] 1 unit upward.
- Thus, we get [tex]\( g(x) = \sqrt[3]{x-2} + 1 \)[/tex].
Combining these two transformations, the function [tex]\( g(x) \)[/tex] is:
[tex]\[ g(x) = \sqrt[3]{x-2} + 1 \][/tex]
So, the correct transformed function is:
[tex]\[ g(x) = \sqrt[3]{x-2} + 1 \][/tex]
