To find the value of [tex]\( g(2) \)[/tex] for the given piecewise function, we need to determine which part of the function to use. The function is defined as:
[tex]\[
g(x) =
\begin{cases}
\left(\frac{1}{2}\right)^{x-3}, & \text{if } x < 2 \\
x^3 - 9x^2 + 27x - 25, & \text{if } x \geq 2
\end{cases}
\][/tex]
Here, we are asked to find [tex]\( g(2) \)[/tex]. Since [tex]\( x = 2 \)[/tex], we use the second part of the piecewise function because [tex]\( 2 \)[/tex] is greater than or equal to [tex]\( 2 \)[/tex]:
[tex]\[
g(x) = x^3 - 9x^2 + 27x - 25
\][/tex]
Substituting [tex]\( x = 2 \)[/tex]:
[tex]\[
g(2) = 2^3 - 9 \cdot 2^2 + 27 \cdot 2 - 25
\][/tex]
First, calculate each term individually:
[tex]\[
2^3 = 8
\][/tex]
[tex]\[
9 \cdot 2^2 = 9 \cdot 4 = 36
\][/tex]
[tex]\[
27 \cdot 2 = 54
\][/tex]
[tex]\[
- 25 \text{ remains as it is}
\][/tex]
Now, combine these results:
[tex]\[
g(2) = 8 - 36 + 54 - 25
\][/tex]
Step-by-step addition and subtraction:
[tex]\[
8 - 36 = -28
\][/tex]
[tex]\[
-28 + 54 = 26
\][/tex]
[tex]\[
26 - 25 = 1
\][/tex]
Therefore, the value of [tex]\( g(2) \)[/tex] is 1.
So, the correct answer is [tex]\( \boxed{1} \)[/tex].
