To evaluate [tex]\( f \circ g\left(\frac{2}{3}\right) \)[/tex], we will follow these steps:
1. Define the function [tex]\( g(x) = \frac{2}{x} \)[/tex]:
The function [tex]\( g \)[/tex] takes an input [tex]\( x \)[/tex] and produces an output by dividing 2 by [tex]\( x \)[/tex].
2. Evaluate [tex]\( g \left( \frac{2}{3} \right) \)[/tex]:
Substitute [tex]\( \frac{2}{3} \)[/tex] for [tex]\( x \)[/tex] in the function [tex]\( g \)[/tex]:
[tex]\[
g\left( \frac{2}{3} \right) = \frac{2}{\frac{2}{3}} = 3
\][/tex]
Therefore, [tex]\( g\left( \frac{2}{3} \right) = 3 \)[/tex].
3. Define the function [tex]\( f(x) \)[/tex]:
Given the problem does not specify [tex]\( f(x) \)[/tex], assume the simplest form which is the identity function [tex]\( f(x) = x \)[/tex].
4. Evaluate [tex]\( f \left( g \left( \frac{2}{3} \right) \right) \)[/tex]:
Using the result from step 2, substitute [tex]\( g\left( \frac{2}{3} \right) = 3 \)[/tex] into the function [tex]\( f \)[/tex]:
[tex]\[
f\left( g\left( \frac{2}{3} \right) \right) = f(3) = 3
\][/tex]
Therefore, the evaluation [tex]\( f \circ g \left( \frac{2}{3} \right) = 3 \)[/tex].
To summarize:
[tex]\[
g \left( \frac{2}{3} \right) = 3 \quad \text{and} \quad f \circ g \left( \frac{2}{3} \right) = 3
\][/tex]
Thus, [tex]\( \boxed{3} \)[/tex] and [tex]\( \boxed{3} \)[/tex].
