Let's determine the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] for the exponential function [tex]\( f(x) = a b^x \)[/tex] that passes through the points [tex]\((0, 2)\)[/tex] and [tex]\((2, 18)\)[/tex].
### Step 1: Find [tex]\(a\)[/tex]
First, we use the point [tex]\((0, 2)\)[/tex]. Plugging this point into the function, we get:
[tex]\[ f(0) = a b^0 \][/tex]
Since [tex]\( b^0 = 1 \)[/tex], this simplifies to:
[tex]\[ f(0) = a \cdot 1 = a \][/tex]
We are given that [tex]\( f(0) = 2 \)[/tex], so:
[tex]\[ a = 2 \][/tex]
### Step 2: Find [tex]\( b \)[/tex]
Next, we use the point [tex]\((2, 18)\)[/tex]. Plugging this point into the function, we get:
[tex]\[ f(2) = a b^2 \][/tex]
We already know that [tex]\( a = 2 \)[/tex]. Substituting [tex]\( a \)[/tex] into the equation, we have:
[tex]\[ 18 = 2 b^2 \][/tex]
To solve for [tex]\( b \)[/tex], we first isolate [tex]\( b^2 \)[/tex]:
[tex]\[ b^2 = \frac{18}{2} = 9 \][/tex]
Now, take the square root of both sides:
[tex]\[ b = \sqrt{9} \][/tex]
So:
[tex]\[ b = 3 \][/tex]
### Conclusion
The values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] for the exponential function are:
[tex]\[
a = 2 \\
b = 3
\][/tex]
