To determine the function [tex]\( g(x) \)[/tex] that is a transformation of the cube root parent function [tex]\( f(x) = \sqrt[3]{x} \)[/tex], let's describe the transformations step by step.
1. Vertical Stretch/Compression:
The factor [tex]\( a \)[/tex] affects vertical stretching or compression. If [tex]\( |a| > 1 \)[/tex], the graph is stretched vertically; if [tex]\( 0 < |a| < 1 \)[/tex], the graph is compressed vertically. For now, we assume [tex]\( a = 1 \)[/tex], meaning there is no vertical stretch or compression.
2. Horizontal Stretch/Compression:
The factor [tex]\( b \)[/tex] affects horizontal stretching or compression. If [tex]\( |b| > 1 \)[/tex], the graph is compressed horizontally; if [tex]\( 0 < |b| < 1 \)[/tex], the graph is stretched horizontally. We assume [tex]\( b = 1 \)[/tex], meaning there is no horizontal stretch or compression.
3. Horizontal Shift:
The parameter [tex]\( h \)[/tex] represents a horizontal shift. If [tex]\( h > 0 \)[/tex], the graph shifts to the right; if [tex]\( h < 0 \)[/tex], the graph shifts to the left. We assume [tex]\( h = 0 \)[/tex], meaning there is no horizontal shift.
4. Vertical Shift:
The parameter [tex]\( k \)[/tex] represents a vertical shift. If [tex]\( k > 0 \)[/tex], the graph shifts up; if [tex]\( k < 0 \)[/tex], the graph shifts down. We assume [tex]\( k = 0 \)[/tex], meaning there is no vertical shift.
In conclusion, we form the function [tex]\( g(x) \)[/tex] from these transformations:
[tex]\[ g(x) = a \cdot \left( b \cdot (x - h) \right)^{1/3} + k \][/tex]
Given our assumptions (identity transformations [tex]\( a = 1 \)[/tex], [tex]\( b = 1 \)[/tex], [tex]\( h = 0 \)[/tex], [tex]\( k = 0 \)[/tex]), we simplify this to:
[tex]\[ g(x) = 1 \cdot \left( 1 \cdot (x - 0) \right)^{1/3} + 0 \][/tex]
Which simplifies to:
[tex]\[ g(x) = \left( x \right)^{1/3} \][/tex]
Hence, the function [tex]\( g(x) \)[/tex] is:
[tex]\[ g(x) = \sqrt[3]{x} \][/tex]
