To solve the problem using the function [tex]\( g(x) = 4(0.6)^x \)[/tex], we will substitute the given values of [tex]\( x \)[/tex] and find the corresponding values of [tex]\( g(x) \)[/tex]. We will also determine the [tex]\( y \)[/tex]-intercept. Let's start with each value step-by-step:
1. For [tex]\( x = -10 \)[/tex]:
[tex]\[ g(-10) = 4(0.6)^{-10} \][/tex]
[tex]\[ g(-10) = 661.5 \][/tex] (rounded to one decimal place)
2. For [tex]\( x = -1 \)[/tex]:
[tex]\[ g(-1) = 4(0.6)^{-1} \][/tex]
[tex]\[ g(-1) = 6.7 \][/tex] (rounded to one decimal place)
3. For [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = 4(0.6)^0 \][/tex]
[tex]\[ g(0) = 4.0 \][/tex] (rounded to one decimal place)
4. For [tex]\( x = 1 \)[/tex]:
[tex]\[ g(1) = 4(0.6)^1 \][/tex]
[tex]\[ g(1) = 2.4 \][/tex] (rounded to one decimal place)
5. For [tex]\( x = 2 \)[/tex]:
[tex]\[ g(2) = 4(0.6)^2 \][/tex]
[tex]\[ g(2) = 1.4 \][/tex] (rounded to one decimal place)
Now we will fill in the table:
\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$g(x)$[/tex] \\
\hline
-10 & 661.5 \\
\hline
-1 & 6.7 \\
\hline
0 & 4.0 \\
\hline
1 & 2.4 \\
\hline
2 & 1.4 \\
\hline
\end{tabular}
The [tex]\( y \)[/tex]-intercept of the function occurs when [tex]\( x = 0 \)[/tex], which gives us the point [tex]\((0, 4.0)\)[/tex].
So, the [tex]\( y \)[/tex]-intercept of the function is at [tex]\( (0, 4.0) \)[/tex].
