To translate the function [tex]\( f(x) = \sqrt[3]{x} \)[/tex]:
1. Translate 3 units in the negative [tex]\( y \)[/tex]-direction:
- Shifting a function [tex]\( k \)[/tex] units in the [tex]\( y \)[/tex]-direction means you subtract [tex]\( k \)[/tex] from the function.
- Here, [tex]\( k = 3 \)[/tex], so the new function is [tex]\( g(x) = \sqrt[3]{x} - 3 \)[/tex].
2. Translate 8 units in the negative [tex]\( x \)[/tex]-direction:
- Shifting a function [tex]\( k \)[/tex] units in the [tex]\( x \)[/tex]-direction means you replace [tex]\( x \)[/tex] with [tex]\( x + k \)[/tex] if the shift is to the left.
- Here, [tex]\( k = -8 \)[/tex] (negative [tex]\( x \)[/tex]-direction), so replace [tex]\( x \)[/tex] with [tex]\( x + 8 \)[/tex] in the function [tex]\( g(x) \)[/tex].
So, the resulting function after both translations is:
[tex]\[ f(x) = \sqrt[3]{x + 8} - 3 \][/tex]
Thus, the correct equation for the resulting function is:
[tex]\[ \boxed{D) \, f(x) = \sqrt[3]{x + 8} - 3} \][/tex]
