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To determine which transformation of \( f(x) = \left(\frac{9}{2}\right)^x \) yields the function \( g(x) = -\left(\frac{9}{2}\right)^{-x} \), let’s analyze the transformations step-by-step:

1. **Reflection across the y-axis**: The expression \( \left(\frac{9}{2}\right)^{-x} \) reflects \( f(x) = \left(\frac{9}{2}\right)^x \) across the y-axis, because replacing \( x \) with \( -x \) inverts the direction of the function.

2. **Reflection across the x-axis**: The negative sign in front of \( \left(\frac{9}{2}\right)^{-x} \) indicates a reflection across the x-axis, flipping the output values.

Thus, applying both a reflection across the y-axis and a reflection across the x-axis to \( f(x) \) results in \( g(x) \). This means the correct answer is:

- **Reflect the graph of \( f \) in the origin**.

Would you like more details, or do you have any further questions?

Here are five related questions for further exploration:

1. How does changing the base of an exponential function affect its graph?
2. What does reflecting a function across the origin mean mathematically?
3. How can transformations be applied to any exponential function to obtain different variations?
4. How do you determine if an exponential function will have a reflection in a certain axis?
5. What other transformations could be applied to an exponential function?

**Tip**: For exponential functions, applying a negative exponent reflects the function across the y-axis, and a negative sign in front of the entire function reflects it across the x-axis.

Describe any transformations of the graph of f(x) = (9/2)^x that yield the graph of g(x) = -(9/2)^(-x).

Explore the transformations needed to convert the graph of f(x) = (9/2)^x to g(x) = -(9/2)^(-x). This problem involves understanding exponential functions, reflection transformations, and their graphical impacts. Suitable for high school students.

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