To determine the function [tex]\( g(x) \)[/tex] after it's transformed from the parent cube root function [tex]\( f(x) = \sqrt[3]{x} \)[/tex], let's consider common transformations such as vertical stretches, reflections, horizontal and vertical translations.
Given the transformations:
- Vertical stretch by a factor of 2: Multiply the function by 2.
- Horizontal shift to the right by 1 unit: Replace [tex]\( x \)[/tex] with [tex]\( x - 1 \)[/tex].
- Vertical shift up by 3 units: Add 3 to the function.
Starting with the parent function:
[tex]\[ f(x) = \sqrt[3]{x} \][/tex]
Let's apply the transformations step by step:
1. Horizontal shift to the right by 1 unit:
- Shift [tex]\( x \)[/tex] to [tex]\( x - 1 \)[/tex]:
[tex]\[ f(x - 1) = \sqrt[3]{x - 1} \][/tex]
2. Vertical stretch by a factor of 2:
- Multiply the function by 2:
[tex]\[ 2 \cdot \sqrt[3]{x - 1} \][/tex]
3. Vertical shift up by 3 units:
- Add 3 to the function:
[tex]\[ 2 \cdot \sqrt[3]{x - 1} + 3 \][/tex]
Therefore, the transformed function [tex]\( g(x) \)[/tex] after applying all the given transformations is:
[tex]\[ g(x) = 2 \cdot \sqrt[3]{x - 1} + 3 \][/tex]
