To find $f(g(x))$ and $g(f(x))$, let's go step by step.

Part (a) Find $f(g(x))$

We are given: $f(x) = x^6 \quad \text{and} \quad g(x) = x^3 - 2x^2 + x - 4$ To find $f(g(x))$, we substitute $g(x)$ into $f(x)$. Since $f(x) = x^6$, this means: $f(g(x)) = (g(x))^6 = (x^3 - 2x^2 + x - 4)^6$ Thus, $f(g(x))$ is: $f(g(x)) = (x^3 - 2x^2 + x - 4)^6$ Expanding this expression fully would give a very large polynomial, but this is the simplified form.

Part (b) Find $g(f(x))$

Now, to find $g(f(x))$, we substitute $f(x)$ into $g(x)$. That is: $g(f(x)) = g(x^6) = (x^6)^3 - 2(x^6)^2 + x^6 - 4$ Simplifying each term: $g(f(x)) = x^{18} - 2x^{12} + x^6 - 4$ So, $g(f(x))$ is: $g(f(x)) = x^{18} - 2x^{12} + x^6 - 4$

Summary:

• $f(g(x)) = (x^3 - 2x^2 + x - 4)^6$
• $g(f(x)) = x^{18} - 2x^{12} + x^6 - 4$

Would you like more details or have any further questions?

Here are 5 related questions:

1. How would you find the derivative of $f(g(x))$?
2. Can $f(g(x))$ be simplified further under specific conditions?
3. How do you find the zeros of $g(f(x))$?
4. What is the general strategy for computing compositions of functions?
5. How do the degrees of $f(g(x))$ and $g(f(x))$ compare?

Tip: When dealing with function compositions, always substitute carefully and watch for opportunities to simplify!