To determine the new function \( g(x) \) after translating the function \( f(x) = x^2 \) 9 units up and 4 units to the right, let's break down the transformations step by step.
1. Translation 4 units to the right:
When a function \( f(x) \) is translated \( c \) units to the right, we replace \( x \) with \( x - c \). In this case, \( c = 4 \). Therefore, translating \( f(x) = x^2 \) 4 units to the right gives:
[tex]\[
f(x - 4) = (x - 4)^2
\][/tex]
2. Translation 9 units up:
When a function is translated \( k \) units up, we add \( k \) to the entire function. In this case, \( k = 9 \). Therefore, translating \( (x - 4)^2 \) 9 units up gives:
[tex]\[
(x - 4)^2 + 9
\][/tex]
Combining both transformations, the resulting function \( g(x) \) is:
[tex]\[
g(x) = (x - 4)^2 + 9
\][/tex]
Therefore, the correct representation of \( g(x) \) is:
[tex]\[
g(x) = (x - 4)^2 + 9
\][/tex]
Hence, the correct option is:
[tex]\[ \boxed{(x - 4)^2 + 9} \][/tex]
