To determine the equation of the exponential function \( f(x) \) from the given table, we need to identify the form of the exponential function:
[tex]\[ f(x) = A \cdot B^x \][/tex]
Here, \( A \) and \( B \) are constants we need to find. The table of values given is:
[tex]\[
\begin{array}{|r|r|}
\hline
x & f(x) \\
\hline
-2 & 32 \\
\hline
-1 & 16 \\
\hline
0 & 8 \\
\hline
1 & 4 \\
\hline
2 & 2 \\
\hline
3 & 1 \\
\hline
\end{array}
\][/tex]
Let's find \( A \) and \( B \) step-by-step:
1. Finding \( A \):
The value \( A \) can be found by looking at \( f(0) \):
[tex]\[
f(0) = A \cdot B^0 = A
\][/tex]
Given \( f(0) = 8 \):
[tex]\[
A = 8
\][/tex]
2. Finding \( B \):
Next, we use two consecutive points from the table to determine \( B \). Let's use \( x = 1 \) and \( x = 0 \) for simplicity:
[tex]\[
f(1) = A \cdot B^1 = 8 \cdot B
\][/tex]
Given \( f(1) = 4 \):
[tex]\[
8 \cdot B = 4
\][/tex]
Solving for \( B \):
[tex]\[
B = \frac{4}{8} = 0.5
\][/tex]
Putting these values of \( A \) and \( B \) together, the equation for the exponential function is:
[tex]\[ f(x) = 8 \cdot (0.5)^x \][/tex]
Therefore, the complete function is:
[tex]\[ f(x) = 8 \cdot 0.5^x \][/tex]
