To determine the function [tex]\( g(x) \)[/tex] as a transformation of the cube root parent function [tex]\( f(x) = \sqrt[3]{x} \)[/tex], let's start by recalling some key concepts about function transformations.
A general transformation of a function [tex]\( f(x) \)[/tex] can be written in the form:
[tex]\[ g(x) = a \cdot f(b(x - h)) + k \][/tex]
where:
- [tex]\( a \)[/tex] is a vertical stretch or compression factor,
- [tex]\( b \)[/tex] is a horizontal stretch or compression factor,
- [tex]\( h \)[/tex] is a horizontal shift,
- [tex]\( k \)[/tex] is a vertical shift.
For the cube root function [tex]\( f(x) = \sqrt[3]{x} \)[/tex]:
1. Vertical Stretch/Compression: If [tex]\( a \neq 1 \)[/tex], the graph of the function will stretch or compress vertically. Here, there is no vertical stretch/compression specified, so [tex]\( a = 1 \)[/tex].
2. Horizontal Stretch/Compression: If [tex]\( b \neq 1 \)[/tex], the graph will stretch or compress horizontally. Here, there is no horizontal stretch/compression specified, so [tex]\( b = 1 \)[/tex].
3. Horizontal Shift: If [tex]\( h \neq 0 \)[/tex], the graph will shift horizontally. Here, no horizontal shift is specified, so [tex]\( h = 0 \)[/tex].
4. Vertical Shift: If [tex]\( k \neq 0 \)[/tex], the graph will shift vertically. Here, no vertical shift is specified, so [tex]\( k = 0 \)[/tex].
Given that [tex]\( a = 1 \)[/tex], [tex]\( b = 1 \)[/tex], [tex]\( h = 0 \)[/tex], and [tex]\( k = 0 \)[/tex], our transformed function [tex]\( g(x) \)[/tex] thus remains the same as [tex]\( f(x) \)[/tex]:
[tex]\[ g(x) = \sqrt[3]{x} \][/tex]
In mathematical terms, this results in the function:
[tex]\[ g(x) = x^{1/3} \][/tex]
Therefore, the function [tex]\( g(x) \)[/tex] is:
[tex]\[ g(x) = x^{1/3} \][/tex]
This completes our step-by-step determination of the transformation, which confirms that [tex]\( g(x) \)[/tex] is indeed [tex]\( x^{1/3} \)[/tex].
