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Consider the table representing an exponential function. The equation for this function is:

[tex]\[
\begin{tabular}{|r|r|}
\hline
[tex]$x$[/tex] & [tex]$f(x)$[/tex] \\
\hline
-2 & 32 \\
\hline
-1 & 16 \\
\hline
0 & 8 \\
\hline
1 & 4 \\
\hline
2 & 2 \\
\hline
3 & 1 \\
\hline
\end{tabular}
\][/tex]

[tex]\[ f(x) = \square (\square)^x \][/tex]

(Note: The placeholders [tex]\(\square\)[/tex] indicate where you need to fill in the coefficients and base for the exponential function.)

Sagot :

GinnyAnsweridn
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]