Given that $(f \circ g)(x) = 8x^3 - 24x^2 + 24x - 8$, we need to determine possible forms for the functions $f(x)$ and $g(x)$.

1. Expressing $(f \circ g)(x)$:

The notation $(f \circ g)(x)$ means that $f(g(x)) = 8x^3 - 24x^2 + 24x - 8$.

To find $f(x)$ and $g(x)$, we will try to express the polynomial $8x^3 - 24x^2 + 24x - 8$ in a way that suggests an inner function $g(x)$ and an outer function $f(x)$.

2. Analyzing the Polynomial:

Let's factor the polynomial $8x^3 - 24x^2 + 24x - 8$: $8x^3 - 24x^2 + 24x - 8 = 8(x^3 - 3x^2 + 3x - 1)$

Next, notice that the cubic expression $x^3 - 3x^2 + 3x - 1$ is of the form $(x-1)^3$. So, we can rewrite the polynomial as: $8x^3 - 24x^2 + 24x - 8 = 8(x - 1)^3$

3. Proposing $f(x)$ and $g(x)$:

Now, we can let:

• $g(x) = x - 1$ (inner function)
• $f(x) = 8x^3$ (outer function)

Then, $(f \circ g)(x) = f(g(x)) = f(x-1) = 8(x-1)^3 = 8x^3 - 24x^2 + 24x - 8$ which matches the given expression.

4. Conclusion:

Thus, the functions $f(x)$ and $g(x)$ could be:

• $f(x) = 8x^3$
• $g(x) = x - 1$

If you need any more clarification or want to explore other possible pairs of $f(x)$ and $g(x)$, feel free to ask!

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Questions:

1. Would you like to explore any other possible forms of $f(x)$ and $g(x)$?
2. Do you need further explanation on the process of factoring the polynomial?
3. Are there any other related functions or compositions you'd like to examine?
4. Would you like to see how to verify the solution by direct substitution?
5. Do you want more practice with compositions of functions and their factorizations?

Tip: When trying to determine functions $f(x)$ and $g(x)$ from a composition, factoring the expression can provide insights into the forms of the inner and outer functions.