Which of the following exponential functions passes through the points left parenthesis 0 comma 12 right parenthesis and left parenthesis 9 comma 96 right parenthesis?

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.

Find the exponential function f(x) = 12 * 2^(x/3) that passes through the points (0, 12) and (9, 96). This problem involves using key algebraic concepts and properties of exponential functions, suitable for students in Grades 9-12.

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