To answer the question of what function [tex]\( g(x) \)[/tex] is, given that it is a transformation of the cube root parent function [tex]\( f(x) = \sqrt[3]{x} \)[/tex], we need to determine the specific transformation applied to [tex]\( f(x) \)[/tex] to produce [tex]\( g(x) \)[/tex].
Step 1: Understand the parent function
The parent function given is:
[tex]\[ f(x) = \sqrt[3]{x} \][/tex]
This is the standard cube root function.
Step 2: Identify the transformation
The transformation mentioned in the question alters the input to the parent function. This typically involves shifting the graph of the function horizontally or vertically. A transformation of the form [tex]\( g(x) = f(x + c) \)[/tex] represents a horizontal shift.
Step 3: Apply the transformation
Here, we need to shift the parent function [tex]\( f(x) = \sqrt[3]{x} \)[/tex] to get the function [tex]\( g(x) \)[/tex]. Specifically, a horizontal shift to the left by 2 units is described by:
[tex]\[ g(x) = f(x + 2) \][/tex]
Step 4: Substitute the transformation into the parent function
Since [tex]\( f(x) = \sqrt[3]{x} \)[/tex], substituting [tex]\( x + 2 \)[/tex] for [tex]\( x \)[/tex] in the parent function gives us:
[tex]\[ g(x) = \sqrt[3]{x + 2} \][/tex]
Thus, the function that represents the transformed cube root function [tex]\( g(x) \)[/tex] is:
[tex]\[ g(x) = \sqrt[3]{x + 2} \][/tex]
Therefore, the function [tex]\( g(x) \)[/tex] is:
[tex]\[ g(x) = \sqrt[3]{x + 2} \][/tex]
