Sure, let's find the translated function step-by-step.
1. Translation 4 units to the right:
- When a function [tex]\(f(x)\)[/tex] is translated [tex]\(c\)[/tex] units to the right, the transformed function becomes [tex]\(f(x - c)\)[/tex].
- For [tex]\(f(x) = x^2\)[/tex] and [tex]\(c = 4\)[/tex], the transformed function is [tex]\(f(x - 4) = (x - 4)^2\)[/tex].
2. Translation 9 units up:
- When a function is then translated [tex]\(d\)[/tex] units upwards, you add [tex]\(d\)[/tex] to the whole function.
- For the function [tex]\((x - 4)^2\)[/tex] and [tex]\(d = 9\)[/tex], the final transformed function is [tex]\((x - 4)^2 + 9\)[/tex].
So, putting it all together:
- Translating [tex]\(f(x) = x^2\)[/tex] 4 units to the right yields [tex]\((x - 4)^2\)[/tex].
- Translating [tex]\((x - 4)^2\)[/tex] 9 units up gives [tex]\((x - 4)^2 + 9\)[/tex].
Thus, the function [tex]\(g(x)\)[/tex] that represents this transformation is:
[tex]\[ g(x) = (x - 4)^2 + 9 \][/tex]
Therefore, the correct choice is:
[tex]\[ \boxed{g(x) = (x - 4)^2 + 9} \][/tex]
