Answer :
To solve for the missing values [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex], we need to evaluate the function [tex]\( g(x) = 4^{\frac{1}{2} x} \)[/tex] at specific values of [tex]\( x \)[/tex].
Step-by-Step Solution:
1. Finding [tex]\( A \)[/tex] when [tex]\( x = -2 \)[/tex]:
[tex]\[ g(-2) = 4^{\frac{1}{2} (-2)} \][/tex]
[tex]\[ = 4^{-1} \][/tex]
[tex]\[ = \frac{1}{4} \][/tex]
2. Finding [tex]\( B \)[/tex] when [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = 4^{\frac{1}{2} (0)} \][/tex]
[tex]\[ = 4^0 \][/tex]
[tex]\[ = 1 \][/tex]
3. Finding [tex]\( C \)[/tex] when [tex]\( x = 2 \)[/tex]:
[tex]\[ g(2) = 4^{\frac{1}{2} (2)} \][/tex]
[tex]\[ = 4^{1} \][/tex]
[tex]\[ = 4 \][/tex]
So, after applying the function [tex]\( g(x) \)[/tex] for each required [tex]\( x \)[/tex], the missing values are:
[tex]\[ \begin{aligned} A &= \frac{1}{4} \\ B &= 1 \\ C &= 4 \end{aligned} \][/tex]
Step-by-Step Solution:
1. Finding [tex]\( A \)[/tex] when [tex]\( x = -2 \)[/tex]:
[tex]\[ g(-2) = 4^{\frac{1}{2} (-2)} \][/tex]
[tex]\[ = 4^{-1} \][/tex]
[tex]\[ = \frac{1}{4} \][/tex]
2. Finding [tex]\( B \)[/tex] when [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = 4^{\frac{1}{2} (0)} \][/tex]
[tex]\[ = 4^0 \][/tex]
[tex]\[ = 1 \][/tex]
3. Finding [tex]\( C \)[/tex] when [tex]\( x = 2 \)[/tex]:
[tex]\[ g(2) = 4^{\frac{1}{2} (2)} \][/tex]
[tex]\[ = 4^{1} \][/tex]
[tex]\[ = 4 \][/tex]
So, after applying the function [tex]\( g(x) \)[/tex] for each required [tex]\( x \)[/tex], the missing values are:
[tex]\[ \begin{aligned} A &= \frac{1}{4} \\ B &= 1 \\ C &= 4 \end{aligned} \][/tex]