To find [tex]\( g(f(0)) \)[/tex], we need to evaluate the function [tex]\( g(f(x)) \)[/tex] at [tex]\( x = 0 \)[/tex].
Given the function:
[tex]\[ g(f(x)) = \frac{4}{9} x^2 - \frac{8}{3} x - 2 \][/tex]
we substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ g(f(0)) = \frac{4}{9} (0)^2 - \frac{8}{3} (0) - 2 \][/tex]
Simplifying each term:
[tex]\[ \frac{4}{9} (0)^2 = \frac{4}{9} \cdot 0 = 0 \][/tex]
[tex]\[ \frac{8}{3} (0) = \frac{8}{3} \cdot 0 = 0 \][/tex]
Thus:
[tex]\[ g(f(0)) = 0 - 0 - 2 \][/tex]
Therefore:
[tex]\[ g(f(0)) = -2 \][/tex]
So, [tex]\( g(f(0)) = -2 \)[/tex].
