To determine the function [tex]\( g(x) \)[/tex] as a transformation of the cube root parent function [tex]\( f(x) = \sqrt[3]{x} \)[/tex], we can analyze the steps involved in transforming the parent function.
1. Shifting the function 1 unit to the left: The transformation required to shift a function [tex]\( c \)[/tex] units to the left can be represented as [tex]\( f(x+c) \)[/tex]. For our problem, [tex]\( c = 1 \)[/tex], so the transformation becomes:
[tex]\[
f(x + 1) = \sqrt[3]{x + 1}
\][/tex]
2. Shifting the function 2 units up: The transformation required to shift a function [tex]\( d \)[/tex] units up can be represented as [tex]\( f(x) + d \)[/tex]. For our problem, [tex]\( d = 2 \)[/tex], so the transformation becomes:
[tex]\[
\sqrt[3]{x + 1} + 2
\][/tex]
Combining these two transformations, we get the function [tex]\( g(x) \)[/tex] as:
[tex]\[
g(x) = \sqrt[3]{x + 1} + 2
\][/tex]
Therefore, the function [tex]\( g(x) \)[/tex] is
[tex]\[
g(x) = \sqrt[3]{x + 1} + 2
\][/tex]
So, the given function matches the transformation described in option A, which is:
[tex]\[
a(x) = \sqrt[3]{x + 1} + 2
\][/tex]
