To solve this problem, we'll analyze the transformations applied to the cube root parent function [tex]\( f(x) = \sqrt[3]{x} \)[/tex].
1. Understanding Transformations:
The general form of a transformed cube root function is:
[tex]\[
g(x) = a \sqrt[3]{x - h} + k
\][/tex]
where [tex]\((h, k)\)[/tex] represents the horizontal and vertical shifts respectively, and [tex]\(a\)[/tex] represents any vertical stretch or compression.
2. Analyzing the Options:
We have two possible functions for [tex]\(g(x)\)[/tex]:
- Option A: [tex]\( g(x) = \sqrt[3]{x - 4} + 3 \)[/tex]
- Option B: [tex]\( g(x) = \sqrt[3]{x - 3} + 4 \)[/tex]
3. Translation Analysis for Option A:
- [tex]\( \sqrt[3]{x - 4} \)[/tex]: The [tex]\( (x - 4) \)[/tex] indicates a horizontal shift 4 units to the right.
- [tex]\( +3 \)[/tex]: This indicates a vertical shift 3 units up.
- So, Option A shifts the parent function [tex]\( \sqrt[3]{x} \)[/tex] 4 units to the right and 3 units up.
4. Translation Analysis for Option B:
- [tex]\( \sqrt[3]{x - 3} \)[/tex]: The [tex]\( (x - 3) \)[/tex] indicates a horizontal shift 3 units to the right.
- [tex]\( +4 \)[/tex]: This indicates a vertical shift 4 units up.
- So, Option B shifts the parent function [tex]\( \sqrt[3]{x} \)[/tex] 3 units to the right and 4 units up.
5. Choosing the Correct Transformation:
To determine which transformation best describes the desired shift, we analyze the provided options and transformations. Upon evaluating both, it's clear that the correct choice is the one that matches the intended shift precisely.
Given our evaluation, the correct transformation of the cube root parent function is:
- Option A: [tex]\( g(x) = \sqrt[3]{x - 4} + 3 \)[/tex]
Therefore, the function [tex]\( g(x) \)[/tex] is [tex]\( \sqrt[3]{x - 4} + 3 \)[/tex].
