We are given the function $f(x) = x^4 - x^2$. The problem asks us to perform two transformations on this function:

1. Horizontal Compression by a Factor of 2: A horizontal compression of a function $f(x)$ by a factor of 2 is represented by replacing $x$ with $\frac{x}{2}$. So, the compressed function becomes: $f\left(\frac{x}{2}\right) = \left(\frac{x}{2}\right)^4 - \left(\frac{x}{2}\right)^2 = \frac{x^4}{16} - \frac{x^2}{4}$

2. Shift 9 Units to the Right: To shift the function $f(x)$ 9 units to the right, replace $x$ with $x - 9$ in the function. Applying this to the compressed function: $f\left(\frac{x - 9}{2}\right) = \left(\frac{x - 9}{2}\right)^4 - \left(\frac{x - 9}{2}\right)^2$ Expanding the terms: $\left(\frac{x - 9}{2}\right)^4 = \frac{(x - 9)^4}{16}, \quad \left(\frac{x - 9}{2}\right)^2 = \frac{(x - 9)^2}{4}$ Therefore, the transformed function is: $f(x) = \frac{(x - 9)^4}{16} - \frac{(x - 9)^2}{4}$

This is the final equation of the transformed function.

Would you like more details on any part of the process?

Related Questions:

1. How do vertical stretches or compressions affect a function?
2. What is the difference between horizontal and vertical transformations?
3. How do translations affect the graph of a function in general?
4. Can you explain how to expand $(x-9)^4$ in detail?
5. What happens to the function if it's shifted vertically instead of horizontally?

Tip:

When dealing with horizontal transformations, always remember that changes inside the function's argument (like $x$) affect the graph in the opposite direction of what you might initially think (compression by 2 means multiplying by $\frac{1}{2}$, and a right shift by 9 means subtracting 9).