To find the correct equation for the given transformations, we need to understand how functions are translated.
Given the original function:
[tex]\[ f(x)=\sqrt[3]{x} \][/tex]
### Step 1: Translate 3 units in the negative [tex]\( y \)[/tex]-direction.
When a function [tex]\( f(x) \)[/tex] is translated [tex]\( k \)[/tex] units in the negative [tex]\( y \)[/tex]-direction, we subtract [tex]\( k \)[/tex] from the function:
[tex]\[ f(x) \to f(x) - 3 \][/tex]
So, the function becomes:
[tex]\[ f(x) - 3 = \sqrt[3]{x} - 3 \][/tex]
### Step 2: Translate 8 units in the negative [tex]\( x \)[/tex]-direction.
When a function [tex]\( f(x) \)[/tex] is translated [tex]\( h \)[/tex] units in the negative [tex]\( x \)[/tex]-direction, we replace [tex]\( x \)[/tex] by [tex]\( x + h \)[/tex]:
[tex]\[ f(x) \to f(x + 8) \][/tex]
So, applying this to our modified function ([tex]\( f(x)-3 \)[/tex]):
[tex]\[ f(x+8) - 3 = \sqrt[3]{x + 8} - 3 \][/tex]
Thus, the correct equation after these two transformations is:
[tex]\[ g(x) = \sqrt[3]{x + 8} - 3 \][/tex]
### Conclusion:
Among the given options:
A) [tex]\( f(x) = \sqrt[3]{x - 3} + 8 \)[/tex]
B) [tex]\( f(x) = \sqrt[3]{x - 8} - 3 \)[/tex]
C) [tex]\( f(x) = \sqrt[3]{x + 3} - 8 \)[/tex]
D) [tex]\( f(x) = \sqrt[3]{x + 8} - 3 \)[/tex]
Option D is the correct equation for the resulting function after translating [tex]\( 3 \)[/tex] units in the negative [tex]\( y \)[/tex]-direction and [tex]\( 8 \)[/tex] units in the negative [tex]\( x \)[/tex]-direction:
[tex]\[ f(x) = \sqrt[3]{x + 8} - 3 \][/tex]
Therefore, the correct choice is:
[tex]\[ \boxed{D} \][/tex]
