If [tex]f(x)[/tex] is an exponential function of the form [tex]y = a b^x[/tex], where [tex]f(-1) = 2[/tex] and [tex]f(2) = 84[/tex], then find the value of [tex]f(2.5)[/tex] to the nearest hundredth.

To solve the problem of finding [tex]$ f(2.5) $[/tex] for the exponential function [tex]$ f(x) = a \cdot b^x $[/tex], given that [tex]$ f(-1) = 2 $[/tex] and [tex]$ f(2) = 84 $[/tex], follow these steps:

1. Set Up Equations:
- From [tex]$ f(-1) = 2 $[/tex], we have the equation:
[tex]\[
a \cdot b^{-1} = 2 \quad \Rightarrow \quad \frac{a}{b} = 2 \quad \Rightarrow \quad a = 2b
\][/tex]

- From [tex]$ f(2) = 84 $[/tex], we get:
[tex]\[
a \cdot b^2 = 84
\][/tex]

2. Substitute [tex]$ a $[/tex]:
- Substitute [tex]$ a = 2b $[/tex] into the second equation:
[tex]\[
(2b) \cdot b^2 = 84 \quad \Rightarrow \quad 2b^3 = 84
\][/tex]

3. Solve for [tex]$ b $[/tex]:
- Divide both sides by 2:
[tex]\[
b^3 = 42
\][/tex]

- Take the cube root to find [tex]$ b $[/tex]:
[tex]\[
b = \sqrt[3]{42} \approx 3.476
\][/tex]

4. Calculate [tex]$ a $[/tex]:
- Since [tex]$ a = 2b $[/tex]:
[tex]\[
a = 2 \cdot 3.476 \approx 6.952
\][/tex]

5. Find [tex]$ f(2.5) $[/tex]:
- Use the function [tex]$ f(x) = a \cdot b^x $[/tex] to find [tex]$ f(2.5) $[/tex]:
[tex]\[
f(2.5) = 6.952 \cdot (3.476)^{2.5} \approx 156.61
\][/tex]

Therefore, the value of [tex]$ f(2.5) $[/tex] is approximately 156.61 to the nearest hundredth.