Let's solve the given problem step by step.
We are given the function [tex]\( f(g(x)) = \frac{4x^2 - 9}{3} \)[/tex] and we need to find [tex]\( f(g(6)) \)[/tex].
First, we need to find the value of [tex]\( g(6) \)[/tex]. Let's assume that the function [tex]\( g(x) \)[/tex] is defined as:
[tex]\[ g(x) = \frac{x^2 - 7x + 10}{2} \][/tex]
We need to substitute [tex]\( x = 6 \)[/tex] in [tex]\( g(x) \)[/tex] to find [tex]\( g(6) \)[/tex]:
[tex]\[
g(6) = \frac{6^2 - 7 \cdot 6 + 10}{2}
\][/tex]
[tex]\[
g(6) = \frac{36 - 42 + 10}{2}
\][/tex]
[tex]\[
g(6) = \frac{4}{2}
\][/tex]
[tex]\[
g(6) = 2
\][/tex]
Now, we plug [tex]\( g(6) \)[/tex] into [tex]\( f(x) \)[/tex] to find [tex]\( f(g(6)) \)[/tex], which means we need to find [tex]\( f(2) \)[/tex] since [tex]\( g(6) = 2 \)[/tex].
So we substitute [tex]\( x = 2 \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = \frac{4x^2 - 9}{3}
\][/tex]
Substitute [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = \frac{4 \cdot 2^2 - 9}{3}
\][/tex]
[tex]\[
f(2) = \frac{4 \cdot 4 - 9}{3}
\][/tex]
[tex]\[
f(2) = \frac{16 - 9}{3}
\][/tex]
[tex]\[
f(2) = \frac{7}{3}
\][/tex]
[tex]\[
f(2) = 2.3333333333333335
\][/tex]
Therefore, [tex]\( f(g(6)) = 2.3333333333333335 \)[/tex].
So the final answer is:
[tex]\[
f(g(6)) = 2.3333333333333335
\][/tex]
