Let's analyze the given options and see how each one transforms the parent function [tex]\( f(x) = \sqrt[3]{x} \)[/tex].
Transformation of functions follows specific rules, and we will apply these rules to the parent function.
1. Option A: [tex]\( g(x) = 3 \sqrt[3]{x} \)[/tex]
- This represents a vertical stretch of the parent function by a factor of 3.
- This transformation implies that every value of [tex]\( \sqrt[3]{x} \)[/tex] is multiplied by 3.
2. Option B: [tex]\( g(x) = \sqrt[3]{x+3} \)[/tex]
- This represents a horizontal shift of the parent function to the left by 3 units.
- In general, [tex]\( \sqrt[3]{x+k} \)[/tex] shifts the graph left by [tex]\( k \)[/tex] units when [tex]\( k \)[/tex] is positive.
3. Option C: [tex]\( g(x) = \frac{1}{3} \sqrt[3]{x} \)[/tex]
- This represents a vertical shrink of the parent function by a factor of [tex]\(\frac{1}{3}\)[/tex].
- This transformation implies that every value of [tex]\( \sqrt[3]{x} \)[/tex] is multiplied by [tex]\(\frac{1}{3}\)[/tex].
4. Option D: [tex]\( g(x) = \sqrt[3]{x} + 3 \)[/tex]
- This represents a vertical shift of the parent function upwards by 3 units.
- In general, [tex]\( \sqrt[3]{x} + k \)[/tex] shifts the graph up by [tex]\( k \)[/tex] units when [tex]\( k \)[/tex] is positive.
Given the function is defined as a transformation of the cube root parent function and we are looking for the function [tex]\( g(x) = \sqrt[3]{x} + 3 \)[/tex], which corresponds to a vertical shift upward by 3 units, the correct transformation is described in Option D.
Thus, the function [tex]\( g(x) \)[/tex] is:
[tex]\[ g(x) = \sqrt[3]{x} + 3 \][/tex]
The function [tex]\( g(x) \)[/tex] you are looking for is [tex]\( g(x) = \sqrt[3]{x} + 3 \)[/tex], and it corresponds to Option D.
