To determine the value of [tex]\( g(2) \)[/tex], we need to apply the definition of the function [tex]\( g(x) \)[/tex] for [tex]\( x = 2 \)[/tex].
Given the piecewise definition of [tex]\( g(x) \)[/tex]:
[tex]\[
g(x)=\left\{
\begin{array}{ll}
\left(\frac{1}{2}\right)^{x-3} & \text{if } x<2 \\
x^3-9x^2+27x-25 & \text{if } x \geq 2
\end{array}
\right.
\][/tex]
Since [tex]\( x = 2 \)[/tex] falls in the category where [tex]\( x \geq 2 \)[/tex], we should use the second part of the piecewise function:
[tex]\[
g(x) = x^3 - 9x^2 + 27x - 25
\][/tex]
Next, substitute [tex]\( x = 2 \)[/tex] into the function:
[tex]\[
g(2) = 2^3 - 9 \cdot 2^2 + 27 \cdot 2 - 25
\][/tex]
Calculating each term individually:
[tex]\[
2^3 = 8
\][/tex]
[tex]\[
9 \cdot 2^2 = 9 \cdot 4 = 36
\][/tex]
[tex]\[
27 \cdot 2 = 54
\][/tex]
So, substituting these values in the expression we get:
[tex]\[
g(2) = 8 - 36 + 54 - 25
\][/tex]
Now, we simplify the expression by performing the arithmetic operations:
[tex]\[
g(2) = 8 - 36 = -28
\][/tex]
[tex]\[
-28 + 54 = 26
\][/tex]
[tex]\[
26 - 25 = 1
\][/tex]
Therefore, the value of [tex]\( g(2) \)[/tex] is:
[tex]\[
\boxed{1}
\][/tex]
Thus, the correct answer is:
[tex]\[
\text{B. } 1
\][/tex]
