To find the function [tex]\( g(x) \)[/tex] given that [tex]\( f(x) = x^2 \)[/tex] has been translated 9 units up and 4 units to the right, we need to apply the translation rules to the function [tex]\( f(x) \)[/tex].
### Horizontal Translation:
When a function [tex]\( f(x) \)[/tex] is translated horizontally by [tex]\( h \)[/tex] units to the right, the new function becomes [tex]\( f(x - h) \)[/tex].
For [tex]\( f(x) = x^2 \)[/tex] translated 4 units to the right, we substitute [tex]\( x \)[/tex] with [tex]\( (x - 4) \)[/tex]:
[tex]\[ f(x - 4) = (x - 4)^2 \][/tex]
### Vertical Translation:
When a function [tex]\( f(x) \)[/tex] is translated vertically by [tex]\( k \)[/tex] units up, the new function becomes [tex]\( f(x) + k \)[/tex].
For [tex]\( f(x) = x^2 \)[/tex] translated 9 units up, we add 9 to the function:
[tex]\[ f(x) + 9 = x^2 + 9 \][/tex]
Combining both translations, we apply the horizontal translation first and then the vertical translation:
[tex]\[ g(x) = (x - 4)^2 + 9 \][/tex]
Thus, the function [tex]\( g(x) \)[/tex] that represents the given transformations is:
[tex]\[ g(x) = (x - 4)^2 + 9 \][/tex]
So the correct answer is:
[tex]\[ g(x) = (x - 4)^2 + 9 \][/tex]
Therefore, the correct option is:
[tex]\[ \boxed{(x-4)^2+9} \][/tex]
