To determine the correct equation for the resulting function when [tex]\( f(x) = \sqrt[3]{x} \)[/tex] is translated, let's analyze the translations step by step.
1. Translation 3 units in the negative [tex]\( y \)[/tex]-direction:
Translating a function down by 3 units means subtracting 3 from the function. So, if the original function is [tex]\( f(x) = \sqrt[3]{x} \)[/tex], after translating down by 3 units, the equation becomes:
[tex]\[
f(x) - 3 = \sqrt[3]{x} - 3
\][/tex]
2. Translation 8 units in the negative [tex]\( x \)[/tex]-direction:
Translating a function to the left by 8 units means replacing [tex]\( x \)[/tex] with [tex]\( x + 8 \)[/tex]. This shifts the graph 8 units to the left. So, the new transformed function after translating by 8 units to the left becomes:
[tex]\[
f(x + 8) - 3
\][/tex]
Considering both translations together, the function [tex]\( f(x) = \sqrt[3]{x} \)[/tex] first moves down by 3 units and then moves 8 units to the left. Therefore, the resulting transformed function is:
[tex]\[
\sqrt[3]{x + 8} - 3
\][/tex]
Hence, the correct equation for the translated function is:
[tex]\[
\boxed{f(x) = \sqrt[3]{x + 8} - 3}
\][/tex]
This corresponds to option A.
