To find 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]\((0, 7000)\)[/tex] and [tex]\((3, 7)\)[/tex], follow these steps:
1. Determine [tex]\( a \)[/tex] using the first point [tex]\((0, 7000)\)[/tex]:
The function [tex]\( f(x) = a b^x \)[/tex] should satisfy [tex]\( f(0) = 7000 \)[/tex].
Substitute [tex]\( x = 0 \)[/tex] and [tex]\( f(0) = 7000 \)[/tex] into the function:
[tex]\[
7000 = a b^0
\][/tex]
Since [tex]\( b^0 = 1 \)[/tex], this simplifies to:
[tex]\[
7000 = a \cdot 1 \implies a = 7000
\][/tex]
2. Determine [tex]\( b \)[/tex] using the second point [tex]\((3, 7)\)[/tex]:
Now use the second point to find [tex]\( b \)[/tex]. Substitute [tex]\( x = 3 \)[/tex] and [tex]\( f(3) = 7 \)[/tex] into the function:
[tex]\[
7 = 7000 \cdot b^3
\][/tex]
We need to solve for [tex]\( b \)[/tex]. Start by isolating [tex]\( b^3 \)[/tex]:
[tex]\[
b^3 = \frac{7}{7000}
\][/tex]
Simplify the fraction:
[tex]\[
b^3 = \frac{1}{1000}
\][/tex]
Now, take the cube root of both sides to solve for [tex]\( b \)[/tex]:
[tex]\[
b = \left(\frac{1}{1000}\right)^{\frac{1}{3}}
\][/tex]
Simplifying the cube root:
[tex]\[
b = 0.1
\][/tex]
So, the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are:
[tex]\[
\begin{array}{l}
a = 7000 \\
b = 0.1
\end{array}
\][/tex]
