We are tasked with finding an exponential function $f(x) = ab^x$ that passes through the points $(0, 12)$ and $(9, 96)$.

Step 1: Use the first point $(0, 12)$

At $x = 0$, the equation becomes: $f(0) = ab^0 = a = 12$ Thus, $a = 12$.

Step 2: Use the second point $(9, 96)$

Substitute $a = 12$, $x = 9$, and $f(9) = 96$ into the equation: $f(9) = 12b^9 = 96$ Now, solve for $b$: $12b^9 = 96 \quad \Rightarrow \quad b^9 = \frac{96}{12} = 8$ $b = \sqrt[9]{8}$ Since $8 = 2^3$, we get: $b = \left( 2^3 \right)^{1/9} = 2^{3/9} = 2^{1/3}$

Thus, $b = 2^{1/3}$.

Final equation

The exponential function is: $f(x) = 12 \cdot \left( 2^{1/3} \right)^x$ or equivalently: $f(x) = 12 \cdot 2^{x/3}$

Conclusion:

The exponential function that passes through the points $(0, 12)$ and $(9, 96)$ is: $f(x) = 12 \cdot 2^{x/3}$

Would you like further details or explanations?

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Here are five related questions for practice:

1. How would the function change if the second point were $(9, 192)$?
2. How do you find the inverse of the exponential function $f(x) = 12 \cdot 2^{x/3}$?
3. What is the significance of the base $b = 2^{1/3}$ in terms of growth rate?
4. How can you verify that the given points satisfy the function $f(x) = 12 \cdot 2^{x/3}$?
5. How would you convert this exponential function into a logarithmic function?

Tip: Always check that your solution satisfies both given points by plugging them into the final equation.