To find the [tex]$y$[/tex]-intercept of the function [tex]$F(x) = 2 \cdot 3^x$[/tex], we need to determine the value of the function when [tex]$x = 0$[/tex]. The [tex]$y$[/tex]-intercept is the point where the graph of the function crosses the [tex]$y$[/tex]-axis.
1. Substitute [tex]$x = 0$[/tex] into the function [tex]$F(x) = 2 \cdot 3^x$[/tex] to find [tex]$F(0)$[/tex].
[tex]\[
F(0) = 2 \cdot 3^0
\][/tex]
2. Simplify the expression. Recall that any number raised to the power of 0 is 1.
[tex]\[
3^0 = 1
\][/tex]
3. Multiply the constant factor 2 with the result from the previous step.
[tex]\[
F(0) = 2 \cdot 1 = 2
\][/tex]
Therefore, the [tex]$y$[/tex]-intercept of the function [tex]$F(x) = 2 \cdot 3^x$[/tex] is at the point [tex]$(0, 2)$[/tex].
So, the correct answer is:
C. [tex]$(0, 2)$[/tex]
