Sure! Let's solve the problem step by step:
### Given Points:
We have the points [tex]\((2, 1)\)[/tex] and [tex]\(\left(3, \frac{1}{3}\right)\)[/tex].
### Step 1: Solve for [tex]\( b \)[/tex]
To find [tex]\( b \)[/tex], we use the relationship between the points given in the form of the exponential function [tex]\( f(x) = a b^x \)[/tex].
For the points [tex]\((2, 1)\)[/tex] and [tex]\(\left(3, \frac{1}{3}\right)\)[/tex], we consider the ratio of the function values:
[tex]\[
\frac{f(3)}{f(2)} = \frac{a b^3}{a b^2} = b
\][/tex]
Plugging in the given function values:
[tex]\[
\frac{\frac{1}{3}}{1} = b
\][/tex]
Thus,
[tex]\[
b = \frac{1}{3}
\][/tex]
### Step 2: Solve for [tex]\( a \)[/tex]
Now, we use one of the given points to solve for [tex]\( a \)[/tex] using the exponential function equation [tex]\( f(x) = a b^x \)[/tex].
Using the point [tex]\((2, 1)\)[/tex]:
[tex]\[
1 = a \left(\frac{1}{3}\right)^2
\][/tex]
This simplifies to:
[tex]\[
1 = a \cdot \frac{1}{9}
\][/tex]
Solving for [tex]\( a \)[/tex]:
[tex]\[
a = 1 \cdot 9 = 9
\][/tex]
### Step 3: Write the Exponential Function
Now that we have both [tex]\( a = 9 \)[/tex] and [tex]\( b = \frac{1}{3} \)[/tex], we can write the exponential function.
Thus, the exponential function is:
[tex]\[
f(x) = 9 \left(\frac{1}{3}\right)^x
\][/tex]
So the exponential function that includes the points [tex]\((2, 1)\)[/tex] and [tex]\(\left(3, \frac{1}{3}\right)\)[/tex] is:
[tex]\[
f(x) = 9 \left(\frac{1}{3}\right)^x
\][/tex]
