The table provided represents values for the exponential function [tex]\( f(x) = 2^x \)[/tex]. Let's analyze how the values in the table are generated step-by-step:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 2^0 = 1
\][/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 2^1 = 2
\][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 2^2 = 4
\][/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[
f(3) = 2^3 = 8
\][/tex]
- For [tex]\( x = 4 \)[/tex]:
[tex]\[
f(4) = 2^4 = 16
\][/tex]
These values correspond exactly to the entries provided in the table:
\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$f(x) = 2^x$[/tex] \\
\hline
0 & 1 \\
\hline
1 & 2 \\
\hline
2 & 4 \\
\hline
3 & 8 \\
\hline
4 & 16 \\
\hline
\end{tabular}
This table helps us understand the behavior of the exponential function [tex]\( f(x) = 2^x \)[/tex]. Each time [tex]\( x \)[/tex] is incremented by 1, the value of [tex]\( f(x) \)[/tex] doubles from the previous value. This is a characteristic property of exponential functions, where the function grows at an increasingly rapid rate as [tex]\( x \)[/tex] increases.
