Given the functions [tex]$f(x)=4^x$[/tex] and [tex]$g(x)=4^{\frac{1}{2} x}$[/tex], use the values provided in the table to determine the missing values.

\begin{tabular}{|c|c|c|}
\hline
[tex]$x$[/tex] & [tex]$f(x)$[/tex] & [tex]$g(x)$[/tex] \\
\hline
-2 & [tex]$\frac{1}{16}$[/tex] & A \\
\hline
-1 & [tex]$\frac{1}{4}$[/tex] & [tex]$\frac{1}{2}$[/tex] \\
\hline
0 & 1 & B \\
\hline
1 & 4 & 2 \\
\hline
2 & 16 & C \\
\hline
\end{tabular}

[tex]\[
\begin{array}{l}
A = \square \\
B = \square \\
C = \square
\end{array}
\][/tex]



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]